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sin(4*a)+cos(2*a)+sin(2*a)*cos(2*a)еслиa=-2 (упростите выражение)

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Примеры

Решение

Вы ввели [src]
sin(4*a) + cos(2*a) + sin(2*a)*cos(2*a)
sin⁡(2a)cos⁡(2a)+sin⁡(4a)+cos⁡(2a)\sin{\left(2 a \right)} \cos{\left(2 a \right)} + \sin{\left(4 a \right)} + \cos{\left(2 a \right)}
Подстановка условия [src]
sin(4*a) + cos(2*a) + sin(2*a)*cos(2*a) при a = -2
подставляем
sin(4*a) + cos(2*a) + sin(2*a)*cos(2*a)
sin⁡(2a)cos⁡(2a)+sin⁡(4a)+cos⁡(2a)\sin{\left(2 a \right)} \cos{\left(2 a \right)} + \sin{\left(4 a \right)} + \cos{\left(2 a \right)}
3*sin(4*a)           
---------- + cos(2*a)
    2
3sin⁡(4a)2+cos⁡(2a)\frac{3 \sin{\left(4 a \right)}}{2} + \cos{\left(2 a \right)}
переменные
a = -2
a=−2a = -2
3*sin(4*(-2))              
------------- + cos(2*(-2))
      2
3sin⁡(4(−2))2+cos⁡(2(−2))\frac{3 \sin{\left(4 (-2) \right)}}{2} + \cos{\left(2 (-2) \right)}
  3*sin(8)         
- -------- + cos(4)
     2
−3sin⁡(8)2+cos⁡(4)- \frac{3 \sin{\left(8 \right)}}{2} + \cos{\left(4 \right)}
Степени [src]
                                              / -2*I*a    2*I*a\                     
                                              |e         e     | /   -2*I*a    2*I*a\
 -2*I*a    2*I*a     /   -4*I*a    4*I*a\   I*|------- + ------|*\- e       + e     /
e         e        I*\- e       + e     /     \   2        2   /                     
------- + ------ - ---------------------- - -----------------------------------------
   2        2                2                                  2
−i(e2ia2+e−2ia2)(e2ia−e−2ia)2−i(e4ia−e−4ia)2+e2ia2+e−2ia2- \frac{i \left(\frac{e^{2 i a}}{2} + \frac{e^{- 2 i a}}{2}\right) \left(e^{2 i a} - e^{- 2 i a}\right)}{2} - \frac{i \left(e^{4 i a} - e^{- 4 i a}\right)}{2} + \frac{e^{2 i a}}{2} + \frac{e^{- 2 i a}}{2}
Численный ответ [src]
cos(2*a)*sin(2*a) + cos(2*a) + sin(4*a)
Рациональный знаменатель [src]
cos(2*a)*sin(2*a) + cos(2*a) + sin(4*a)
sin⁡(2a)cos⁡(2a)+sin⁡(4a)+cos⁡(2a)\sin{\left (2 a \right )} \cos{\left (2 a \right )} + \sin{\left (4 a \right )} + \cos{\left (2 a \right )}
Объединение рациональных выражений [src]
cos(2*a)*sin(2*a) + cos(2*a) + sin(4*a)
sin⁡(2a)cos⁡(2a)+sin⁡(4a)+cos⁡(2a)\sin{\left (2 a \right )} \cos{\left (2 a \right )} + \sin{\left (4 a \right )} + \cos{\left (2 a \right )}
Общее упрощение [src]
3*sin(4*a)           
---------- + cos(2*a)
    2
3sin⁡(4a)2+cos⁡(2a)\frac{3 \sin{\left(4 a \right)}}{2} + \cos{\left(2 a \right)}
Собрать выражение [src]
sin(2*a)*cos(2*a) + cos(2*a) + sin(4*a)
sin⁡(2a)cos⁡(2a)+sin⁡(4a)+cos⁡(2a)\sin{\left (2 a \right )} \cos{\left (2 a \right )} + \sin{\left (4 a \right )} + \cos{\left (2 a \right )}
Общий знаменатель [src]
cos(2*a)*sin(2*a) + cos(2*a) + sin(4*a)
sin⁡(2a)cos⁡(2a)+sin⁡(4a)+cos⁡(2a)\sin{\left (2 a \right )} \cos{\left (2 a \right )} + \sin{\left (4 a \right )} + \cos{\left (2 a \right )}
Тригонометрическая часть [src]
       2                   
1 - tan (a)     3*tan(2*a) 
----------- + -------------
       2             2     
1 + tan (a)   1 + tan (2*a)
1−tan⁡2(a)tan⁡2(a)+1+3tan⁡(2a)tan⁡2(2a)+1\frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} + \frac{3 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1}
//   0      for And(im(a) = 0, 2*a mod pi = 0)\ //   1      for And(im(a) = 0, a mod pi = 0)\   //   0      for And(im(a) = 0, 4*a mod pi = 0)\   //   1      for And(im(a) = 0, a mod pi = 0)\
|<                                            |*|<                                          | + |<                                            | + |<                                          |
\\sin(2*a)              otherwise             / \\cos(2*a)             otherwise            /   \\sin(4*a)              otherwise             /   \\cos(2*a)             otherwise            /
(({0for im⁡(a)=0∧2a mod π=0sin⁡(2a)otherwise)({1for im⁡(a)=0∧a mod π=0cos⁡(2a)otherwise))+({0for im⁡(a)=0∧4a mod π=0sin⁡(4a)otherwise)+({1for im⁡(a)=0∧a mod π=0cos⁡(2a)otherwise)\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)
   1             1                   1           
-------- + ------------- + ----------------------
csc(4*a)      /pi      \               /pi      \
           csc|-- - 2*a|   csc(2*a)*csc|-- - 2*a|
              \2       /               \2       /
1csc⁡(−2a+π2)+1csc⁡(4a)+1csc⁡(2a)csc⁡(−2a+π2)\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(4 a \right)}} + \frac{1}{\csc{\left(2 a \right)} \csc{\left(- 2 a + \frac{\pi}{2} \right)}}
   1             1                   1           
-------- + ------------- + ----------------------
sec(2*a)      /      pi\               /      pi\
           sec|4*a - --|   sec(2*a)*sec|2*a - --|
              \      2 /               \      2 /
1sec⁡(4a−π2)+1sec⁡(2a)+1sec⁡(2a)sec⁡(2a−π2)\frac{1}{\sec{\left(4 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 a \right)}} + \frac{1}{\sec{\left(2 a \right)} \sec{\left(2 a - \frac{\pi}{2} \right)}}
//      0        for And(im(a) = 0, 2*a mod pi = 0)\                                                 //      0        for And(im(a) = 0, 4*a mod pi = 0)\                                                
||                                                 | //   1      for And(im(a) = 0, a mod pi = 0)\   ||                                                 |   //   1      for And(im(a) = 0, a mod pi = 0)\
|<   /      pi\                                    |*|<                                          | + |<   /      pi\                                    | + |<                                          |
||cos|2*a - --|              otherwise             | \\cos(2*a)             otherwise            /   ||cos|4*a - --|              otherwise             |   \\cos(2*a)             otherwise            /
\\   \      2 /                                    /                                                 \\   \      2 /                                    /
(({0for im⁡(a)=0∧2a mod π=0cos⁡(2a−π2)otherwise)({1for im⁡(a)=0∧a mod π=0cos⁡(2a)otherwise))+({0for im⁡(a)=0∧4a mod π=0cos⁡(4a−π2)otherwise)+({1for im⁡(a)=0∧a mod π=0cos⁡(2a)otherwise)\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\cos{\left(2 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\cos{\left(4 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)
            /pi      \                 /pi      \
sin(2*a)*sin|-- + 2*a| + sin(4*a) + sin|-- + 2*a|
            \2       /                 \2       /
sin⁡(2a)sin⁡(2a+π2)+sin⁡(4a)+sin⁡(2a+π2)\sin{\left(2 a \right)} \sin{\left(2 a + \frac{\pi}{2} \right)} + \sin{\left(4 a \right)} + \sin{\left(2 a + \frac{\pi}{2} \right)}
3*sin(4*a)      /pi      \
---------- + sin|-- + 2*a|
    2           \2       /
3sin⁡(4a)2+sin⁡(2a+π2)\frac{3 \sin{\left(4 a \right)}}{2} + \sin{\left(2 a + \frac{\pi}{2} \right)}
   1          1               1        
-------- + -------- + -----------------
csc(4*a)   sec(2*a)   csc(2*a)*sec(2*a)
1sec⁡(2a)+1csc⁡(4a)+1csc⁡(2a)sec⁡(2a)\frac{1}{\sec{\left(2 a \right)}} + \frac{1}{\csc{\left(4 a \right)}} + \frac{1}{\csc{\left(2 a \right)} \sec{\left(2 a \right)}}
  //      0        for And(im(a) = 0, 4*a mod pi = 0)\                                                    
  ||                                                 |                                                    
  ||  2*cot(2*a)                                     |                                                    
3*|<-------------              otherwise             |                                                    
  ||       2                                         |   //     1        for And(im(a) = 0, a mod pi = 0)\
  ||1 + cot (2*a)                                    |   ||                                              |
  \\                                                 /   ||        2                                     |
------------------------------------------------------ + |<-1 + cot (a)                                  |
                          2                              ||------------             otherwise            |
                                                         ||       2                                      |
                                                         \\1 + cot (a)                                   /
(3({0for im⁡(a)=0∧4a mod π=02cot⁡(2a)cot⁡2(2a)+1otherwise)2)+({1for im⁡(a)=0∧a mod π=0cot⁡2(a)−1cot⁡2(a)+1otherwise)\left(\frac{3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right)}{2}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)
     /      pi\           
3*cos|4*a - --|           
     \      2 /           
--------------- + cos(2*a)
       2
cos⁡(2a)+3cos⁡(4a−π2)2\cos{\left(2 a \right)} + \frac{3 \cos{\left(4 a - \frac{\pi}{2} \right)}}{2}
3*sin(4*a)           
---------- + cos(2*a)
    2
3sin⁡(4a)2+cos⁡(2a)\frac{3 \sin{\left(4 a \right)}}{2} + \cos{\left(2 a \right)}
                                                //      1        for And(im(a) = 0, a mod pi = 0)\                                                     //      1        for And(im(a) = 0, a mod pi = 0)\
//   0      for And(im(a) = 0, 2*a mod pi = 0)\ ||                                               |   //   0      for And(im(a) = 0, 4*a mod pi = 0)\   ||                                               |
|<                                            |*|<   /pi      \                                  | + |<                                            | + |<   /pi      \                                  |
\\sin(2*a)              otherwise             / ||sin|-- + 2*a|             otherwise            |   \\sin(4*a)              otherwise             /   ||sin|-- + 2*a|             otherwise            |
                                                \\   \2       /                                  /                                                     \\   \2       /                                  /
(({0for im⁡(a)=0∧2a mod π=0sin⁡(2a)otherwise)({1for im⁡(a)=0∧a mod π=0sin⁡(2a+π2)otherwise))+({0for im⁡(a)=0∧4a mod π=0sin⁡(4a)otherwise)+({1for im⁡(a)=0∧a mod π=0sin⁡(2a+π2)otherwise)\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(2 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(2 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)
   1              3       
-------- + ---------------
sec(2*a)        /      pi\
           2*sec|4*a - --|
                \      2 /
32sec⁡(4a−π2)+1sec⁡(2a)\frac{3}{2 \sec{\left(4 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 a \right)}}
                                                //      1        for And(im(a) = 0, a mod pi = 0)\                                                     //      1        for And(im(a) = 0, a mod pi = 0)\
//   0      for And(im(a) = 0, 2*a mod pi = 0)\ ||                                               |   //   0      for And(im(a) = 0, 4*a mod pi = 0)\   ||                                               |
||                                            | ||      1                                        |   ||                                            |   ||      1                                        |
|<   1                                        |*|<-------------             otherwise            | + |<   1                                        | + |<-------------             otherwise            |
||--------              otherwise             | ||   /pi      \                                  |   ||--------              otherwise             |   ||   /pi      \                                  |
\\csc(2*a)                                    / ||csc|-- - 2*a|                                  |   \\csc(4*a)                                    /   ||csc|-- - 2*a|                                  |
                                                \\   \2       /                                  /                                                     \\   \2       /                                  /
(({0for im⁡(a)=0∧2a mod π=01csc⁡(2a)otherwise)({1for im⁡(a)=0∧a mod π=01csc⁡(−2a+π2)otherwise))+({0for im⁡(a)=0∧4a mod π=01csc⁡(4a)otherwise)+({1for im⁡(a)=0∧a mod π=01csc⁡(−2a+π2)otherwise)\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{1}{\csc{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\frac{1}{\csc{\left(4 a \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)
      1             3     
------------- + ----------
   /pi      \   2*csc(4*a)
csc|-- - 2*a|             
   \2       /
1csc⁡(−2a+π2)+32csc⁡(4a)\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{3}{2 \csc{\left(4 a \right)}}
            /      pi\                 /      pi\
cos(2*a)*cos|2*a - --| + cos(2*a) + cos|4*a - --|
            \      2 /                 \      2 /
cos⁡(2a)cos⁡(2a−π2)+cos⁡(2a)+cos⁡(4a−π2)\cos{\left(2 a \right)} \cos{\left(2 a - \frac{\pi}{2} \right)} + \cos{\left(2 a \right)} + \cos{\left(4 a - \frac{\pi}{2} \right)}
//     0       for And(im(a) = 0, 2*a mod pi = 0)\ //     1       for And(im(a) = 0, a mod pi = 0)\   //      0        for And(im(a) = 0, 4*a mod pi = 0)\   //     1       for And(im(a) = 0, a mod pi = 0)\
||                                               | ||                                             |   ||                                                 |   ||                                             |
||  2*tan(a)                                     | ||       2                                     |   ||  2*tan(2*a)                                     |   ||       2                                     |
|<-----------              otherwise             |*|<1 - tan (a)                                  | + |<-------------              otherwise             | + |<1 - tan (a)                                  |
||       2                                       | ||-----------             otherwise            |   ||       2                                         |   ||-----------             otherwise            |
||1 + tan (a)                                    | ||       2                                     |   ||1 + tan (2*a)                                    |   ||       2                                     |
\\                                               / \\1 + tan (a)                                  /   \\                                                 /   \\1 + tan (a)                                  /
(({0for im⁡(a)=0∧2a mod π=02tan⁡(a)tan⁡2(a)+1otherwise)({1for im⁡(a)=0∧a mod π=01−tan⁡2(a)tan⁡2(a)+1otherwise))+({0for im⁡(a)=0∧4a mod π=02tan⁡(2a)tan⁡2(2a)+1otherwise)+({1for im⁡(a)=0∧a mod π=01−tan⁡2(a)tan⁡2(a)+1otherwise)\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{2 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\frac{2 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)
//     0       for And(im(a) = 0, 2*a mod pi = 0)\ //     1        for And(im(a) = 0, a mod pi = 0)\   //      0        for And(im(a) = 0, 4*a mod pi = 0)\   //     1        for And(im(a) = 0, a mod pi = 0)\
||                                               | ||                                              |   ||                                                 |   ||                                              |
||  2*cot(a)                                     | ||        2                                     |   ||  2*cot(2*a)                                     |   ||        2                                     |
|<-----------              otherwise             |*|<-1 + cot (a)                                  | + |<-------------              otherwise             | + |<-1 + cot (a)                                  |
||       2                                       | ||------------             otherwise            |   ||       2                                         |   ||------------             otherwise            |
||1 + cot (a)                                    | ||       2                                      |   ||1 + cot (2*a)                                    |   ||       2                                      |
\\                                               / \\1 + cot (a)                                   /   \\                                                 /   \\1 + cot (a)                                   /
(({0for im⁡(a)=0∧2a mod π=02cot⁡(a)cot⁡2(a)+1otherwise)({1for im⁡(a)=0∧a mod π=0cot⁡2(a)−1cot⁡2(a)+1otherwise))+({0for im⁡(a)=0∧4a mod π=02cot⁡(2a)cot⁡2(2a)+1otherwise)+({1for im⁡(a)=0∧a mod π=0cot⁡2(a)−1cot⁡2(a)+1otherwise)\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)
//                      0                        for And(im(a) = 0, 2*a mod pi = 0)\ //                     1                       for And(im(a) = 0, a mod pi = 0)\   //                      0                        for And(im(a) = 0, 4*a mod pi = 0)\   //                     1                       for And(im(a) = 0, a mod pi = 0)\
||                                                                                 | ||                                                                             |   ||                                                                                 |   ||                                                                             |
|
(({0for im⁡(a)=0∧2a mod π=0{0for im⁡(a)=0∧2a mod π=0sin⁡(2a)otherwiseotherwise)({1for im⁡(a)=0∧a mod π=0{1for im⁡(a)=0∧a mod π=0cos⁡(2a)otherwiseotherwise))+({0for im⁡(a)=0∧4a mod π=0{0for im⁡(a)=0∧4a mod π=0sin⁡(4a)otherwiseotherwise)+({1for im⁡(a)=0∧a mod π=0{1for im⁡(a)=0∧a mod π=0cos⁡(2a)otherwiseotherwise)\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)
       2                        /       2   \       
1 - tan (a)     2*tan(2*a)    2*\1 - tan (a)/*tan(a)
----------- + ------------- + ----------------------
       2             2                         2    
1 + tan (a)   1 + tan (2*a)       /       2   \     
                                  \1 + tan (a)/
1−tan⁡2(a)tan⁡2(a)+1+2⋅(1−tan⁡2(a))tan⁡(a)(tan⁡2(a)+1)2+2tan⁡(2a)tan⁡2(2a)+1\frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} + \frac{2 \cdot \left(1 - \tan^{2}{\left(a \right)}\right) \tan{\left(a \right)}}{\left(\tan^{2}{\left(a \right)} + 1\right)^{2}} + \frac{2 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1}
//                       0                          for And(im(a) = 0, 2*a mod pi = 0)\ //                       1                         for And(im(a) = 0, a mod pi = 0)\   //                        0                           for And(im(a) = 0, 4*a mod pi = 0)\   //                       1                         for And(im(a) = 0, a mod pi = 0)\
||                                                                                    | ||                                                                                 |   ||                                                                                      |   ||                                                                                 |
||/     0       for And(im(a) = 0, 2*a mod pi = 0)                                    | ||/     1        for And(im(a) = 0, a mod pi = 0)                                  |   ||/      0        for And(im(a) = 0, 4*a mod pi = 0)                                    |   ||/     1        for And(im(a) = 0, a mod pi = 0)                                  |
|||                                                                                   | |||                                                                                |   |||                                                                                     |   |||                                                                                |
|<|  2*cot(a)                                                                         |*|<|        2                                                                       | + |<|  2*cot(2*a)                                                                         | + |<|        2                                                                       |
||<-----------              otherwise                           otherwise             | ||<-1 + cot (a)                                               otherwise            |   ||<-------------              otherwise                           otherwise             |   ||<-1 + cot (a)                                               otherwise            |
|||       2                                                                           | |||------------             otherwise                                              |   |||       2                                                                             |   |||------------             otherwise                                              |
|||1 + cot (a)                                                                        | |||       2                                                                        |   |||1 + cot (2*a)                                                                        |   |||       2                                                                        |
\\\                                                                                   / \\\1 + cot (a)                                                                     /   \\\                                                                                     /   \\\1 + cot (a)                                                                     /
(({0for im⁡(a)=0∧2a mod π=0{0for im⁡(a)=0∧2a mod π=02cot⁡(a)cot⁡2(a)+1otherwiseotherwise)({1for im⁡(a)=0∧a mod π=0{1for im⁡(a)=0∧a mod π=0cot⁡2(a)−1cot⁡2(a)+1otherwiseotherwise))+({0for im⁡(a)=0∧4a mod π=0{0for im⁡(a)=0∧4a mod π=02cot⁡(2a)cot⁡2(2a)+1otherwiseotherwise)+({1for im⁡(a)=0∧a mod π=0{1for im⁡(a)=0∧a mod π=0cot⁡2(a)−1cot⁡2(a)+1otherwiseotherwise)\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)
//      0        for And(im(a) = 0, 2*a mod pi = 0)\                                                 //      0        for And(im(a) = 0, 4*a mod pi = 0)\                                                
||                                                 | //   1      for And(im(a) = 0, a mod pi = 0)\   ||                                                 |   //   1      for And(im(a) = 0, a mod pi = 0)\
||      1                                          | ||                                          |   ||      1                                          |   ||                                          |
|<-------------              otherwise             |*|<   1                                      | + |<-------------              otherwise             | + |<   1                                      |
||   /      pi\                                    | ||--------             otherwise            |   ||   /      pi\                                    |   ||--------             otherwise            |
||sec|2*a - --|                                    | \\sec(2*a)                                  /   ||sec|4*a - --|                                    |   \\sec(2*a)                                  /
\\   \      2 /                                    /                                                 \\   \      2 /                                    /
(({0for im⁡(a)=0∧2a mod π=01sec⁡(2a−π2)otherwise)({1for im⁡(a)=0∧a mod π=01sec⁡(2a)otherwise))+({0for im⁡(a)=0∧4a mod π=01sec⁡(4a−π2)otherwise)+({1for im⁡(a)=0∧a mod π=01sec⁡(2a)otherwise)\left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\frac{1}{\sec{\left(4 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right)
  //   0      for And(im(a) = 0, 4*a mod pi = 0)\                                                
3*|<                                            |                                                
  \\sin(4*a)              otherwise             /   //   1      for And(im(a) = 0, a mod pi = 0)\
------------------------------------------------- + |<                                          |
                        2                           \\cos(2*a)             otherwise            /
(3({0for im⁡(a)=0∧4a mod π=0sin⁡(4a)otherwise)2)+({1for im⁡(a)=0∧a mod π=0cos⁡(2a)otherwise)\left(\frac{3 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right)}{2}\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)
Комбинаторика [src]
cos(2*a)*sin(2*a) + cos(2*a) + sin(4*a)
sin⁡(2a)cos⁡(2a)+sin⁡(4a)+cos⁡(2a)\sin{\left (2 a \right )} \cos{\left (2 a \right )} + \sin{\left (4 a \right )} + \cos{\left (2 a \right )}
Раскрыть выражение [src]
          2           3                                    3          
-1 + 2*cos (a) - 8*sin (a)*cos(a) + 2*cos(a)*sin(a) + 4*cos (a)*sin(a)
−8sin⁡3(a)cos⁡(a)+4sin⁡(a)cos⁡3(a)+2sin⁡(a)cos⁡(a)+2cos⁡2(a)−1- 8 \sin^{3}{\left(a \right)} \cos{\left(a \right)} + 4 \sin{\left(a \right)} \cos^{3}{\left(a \right)} + 2 \sin{\left(a \right)} \cos{\left(a \right)} + 2 \cos^{2}{\left(a \right)} - 1