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cos(2*a)*cos(3*a)-sin(2*a)*sin(3*a)еслиa=3 (упростите выражение)

Учитель очень удивится увидев твоё верное решение 😼

Примеры

Решение

Вы ввели [src]
cos(2*a)*cos(3*a) - sin(2*a)*sin(3*a)
−sin⁡(2a)sin⁡(3a)+cos⁡(2a)cos⁡(3a)- \sin{\left(2 a \right)} \sin{\left(3 a \right)} + \cos{\left(2 a \right)} \cos{\left(3 a \right)}
Подстановка условия [src]
cos(2*a)*cos(3*a) - sin(2*a)*sin(3*a) при a = 3
подставляем
cos(2*a)*cos(3*a) - sin(2*a)*sin(3*a)
−sin⁡(2a)sin⁡(3a)+cos⁡(2a)cos⁡(3a)- \sin{\left(2 a \right)} \sin{\left(3 a \right)} + \cos{\left(2 a \right)} \cos{\left(3 a \right)}
cos(5*a)
cos⁡(5a)\cos{\left(5 a \right)}
переменные
a = 3
a=3a = 3
cos(5*(3))
cos⁡(5(3))\cos{\left(5 (3) \right)}
cos(5*3)
cos⁡(5⋅3)\cos{\left(5 \cdot 3 \right)}
cos(15)
cos⁡(15)\cos{\left(15 \right)}
Степени [src]
/ -3*I*a    3*I*a\ / -2*I*a    2*I*a\   /   -3*I*a    3*I*a\ /   -2*I*a    2*I*a\
|e         e     | |e         e     |   \- e       + e     /*\- e       + e     /
|------- + ------|*|------- + ------| + -----------------------------------------
\   2        2   / \   2        2   /                       4
(e2ia2+e−2ia2)(e3ia2+e−3ia2)+(e2ia−e−2ia)(e3ia−e−3ia)4\left(\frac{e^{2 i a}}{2} + \frac{e^{- 2 i a}}{2}\right) \left(\frac{e^{3 i a}}{2} + \frac{e^{- 3 i a}}{2}\right) + \frac{\left(e^{2 i a} - e^{- 2 i a}\right) \left(e^{3 i a} - e^{- 3 i a}\right)}{4}
Численный ответ [src]
cos(2*a)*cos(3*a) - sin(2*a)*sin(3*a)
Общее упрощение [src]
cos(5*a)
cos⁡(5a)\cos{\left(5 a \right)}
Собрать выражение [src]
cos(5*a)
cos⁡(5a)\cos{\left (5 a \right )}
Тригонометрическая часть [src]
/       2   \ /       2/3*a\\                     /3*a\      
\1 - tan (a)/*|1 - tan |---||         4*tan(a)*tan|---|      
              \        \ 2 //                     \ 2 /      
----------------------------- - -----------------------------
/       2   \ /       2/3*a\\   /       2   \ /       2/3*a\\
\1 + tan (a)/*|1 + tan |---||   \1 + tan (a)/*|1 + tan |---||
              \        \ 2 //                 \        \ 2 //
(1−tan⁡2(a))(1−tan⁡2(3a2))(tan⁡2(a)+1)(tan⁡2(3a2)+1)−4tan⁡(a)tan⁡(3a2)(tan⁡2(a)+1)(tan⁡2(3a2)+1)\frac{\left(1 - \tan^{2}{\left(a \right)}\right) \left(1 - \tan^{2}{\left(\frac{3 a}{2} \right)}\right)}{\left(\tan^{2}{\left(a \right)} + 1\right) \left(\tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right)} - \frac{4 \tan{\left(a \right)} \tan{\left(\frac{3 a}{2} \right)}}{\left(\tan^{2}{\left(a \right)} + 1\right) \left(\tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right)}
       2/5*a\
1 - tan |---|
        \ 2 /
-------------
       2/5*a\
1 + tan |---|
        \ 2 /
1−tan⁡2(5a2)tan⁡2(5a2)+1\frac{1 - \tan^{2}{\left(\frac{5 a}{2} \right)}}{\tan^{2}{\left(\frac{5 a}{2} \right)} + 1}
                                                 //      1        for And(im(a) = 0, 3*a mod 2*pi = 0)\                                                      //      0        for And(im(a) = 0, 3*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)\ ||                                                 |
||                                             | ||       2/3*a\                                      |   ||                                               | ||       /3*a\                                     |
||       2                                     | ||1 - tan |---|                                      |   ||  2*tan(a)                                     | ||  2*tan|---|                                     |
|<1 - tan (a)                                  |*|<        \ 2 /                                      | - |<-----------              otherwise             |*|<       \ 2 /                                     |
||-----------             otherwise            | ||-------------               otherwise              |   ||       2                                       | ||-------------              otherwise             |
||       2                                     | ||       2/3*a\                                      |   ||1 + tan (a)                                    | ||       2/3*a\                                    |
\\1 + tan (a)                                  / ||1 + tan |---|                                      |   \\                                               / ||1 + tan |---|                                    |
                                                 \\        \ 2 /                                      /                                                      \\        \ 2 /                                    /
(−({0for im⁡(a)=0∧2a mod π=02tan⁡(a)tan⁡2(a)+1otherwise)({0for im⁡(a)=0∧3a mod π=02tan⁡(3a2)tan⁡2(3a2)+1otherwise))+(({1for im⁡(a)=0∧a mod π=01−tan⁡2(a)tan⁡2(a)+1otherwise)({1for im⁡(a)=0∧3a mod 2π=01−tan⁡2(3a2)tan⁡2(3a2)+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} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\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) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)
                                                                                     //                          1                            for And(im(a) = 0, 3*a mod 2*pi = 0)\                                                                                           //                        0                           for And(im(a) = 0, 3*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, 3*a mod 2*pi = 0)                                      |   ||                                                                                    | ||/      0        for And(im(a) = 0, 3*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)                                    | |||                                                                                     |
|||                                                                                | |||        2/3*a\                                                                            |   |||                                                                                   | |||       /3*a\                                                                         |
|<|        2                                                                       |*|<|-1 + cot |---|                                                                            | - |<|  2*cot(a)                                                                         |*|<|  2*cot|---|                                                                         |
||<-1 + cot (a)                                               otherwise            | ||<         \ 2 /                                                     otherwise              |   ||<-----------              otherwise                           otherwise             | ||<       \ 2 /                                                   otherwise             |
|||------------             otherwise                                              | |||--------------               otherwise                                                    |   |||       2                                                                           | |||-------------              otherwise                                                 |
|||       2                                                                        | |||       2/3*a\                                                                             |   |||1 + cot (a)                                                                        | |||       2/3*a\                                                                        |
\\\1 + cot (a)                                                                     / |||1 + cot |---|                                                                             |   \\\                                                                                   / |||1 + cot |---|                                                                        |
                                                                                     \\\        \ 2 /                                                                             /                                                                                           \\\        \ 2 /                                                                        /
(−({0for im⁡(a)=0∧2a mod π=0{0for im⁡(a)=0∧2a mod π=02cot⁡(a)cot⁡2(a)+1otherwiseotherwise)({0for im⁡(a)=0∧3a mod π=0{0for im⁡(a)=0∧3a mod π=02cot⁡(3a2)cot⁡2(3a2)+1otherwiseotherwise))+(({1for im⁡(a)=0∧a mod π=0{1for im⁡(a)=0∧a mod π=0cot⁡2(a)−1cot⁡2(a)+1otherwiseotherwise)({1for im⁡(a)=0∧3a mod 2π=0{1for im⁡(a)=0∧3a mod 2π=0cot⁡2(3a2)−1cot⁡2(3a2)+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} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\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) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)
      1      
-------------
   /pi      \
csc|-- - 5*a|
   \2       /
1csc⁡(−5a+π2)\frac{1}{\csc{\left(- 5 a + \frac{\pi}{2} \right)}}
                                                                                                  //      0        for And(im(a) = 0, 2*a mod pi = 0)\ //      0        for And(im(a) = 0, 3*a mod pi = 0)\
//   1      for And(im(a) = 0, a mod pi = 0)\ //   1      for And(im(a) = 0, 3*a mod 2*pi = 0)\   ||                                                 | ||                                                 |
|<                                          |*|<                                              | - |<   /      pi\                                    |*|<   /      pi\                                    |
\\cos(2*a)             otherwise            / \\cos(3*a)               otherwise              /   ||cos|2*a - --|              otherwise             | ||cos|3*a - --|              otherwise             |
                                                                                                  \\   \      2 /                                    / \\   \      2 /                                    /
(−({0for im⁡(a)=0∧2a mod π=0cos⁡(2a−π2)otherwise)({0for im⁡(a)=0∧3a mod π=0cos⁡(3a−π2)otherwise))+(({1for im⁡(a)=0∧a mod π=0cos⁡(2a)otherwise)({1for im⁡(a)=0∧3a mod 2π=0cos⁡(3a)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} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\cos{\left(3 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\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) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right)
   1    
--------
sec(5*a)
1sec⁡(5a)\frac{1}{\sec{\left(5 a \right)}}
//   1      for And(im(a) = 0, a mod pi = 0)\ //   1      for And(im(a) = 0, 3*a mod 2*pi = 0)\   //   0      for And(im(a) = 0, 2*a mod pi = 0)\ //   0      for And(im(a) = 0, 3*a mod pi = 0)\
|<                                          |*|<                                              | - |<                                            |*|<                                            |
\\cos(2*a)             otherwise            / \\cos(3*a)               otherwise              /   \\sin(2*a)              otherwise             / \\sin(3*a)              otherwise             /
(−({0for im⁡(a)=0∧2a mod π=0sin⁡(2a)otherwise)({0for im⁡(a)=0∧3a mod π=0sin⁡(3a)otherwise))+(({1for im⁡(a)=0∧a mod π=0cos⁡(2a)otherwise)({1for im⁡(a)=0∧3a mod 2π=0cos⁡(3a)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} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\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) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right)
/      1         for And(im(a) = 0, 5*a mod 2*pi = 0)
|                                                    
|        2/5*a\                                      
|-1 + cot |---|                                      
<         \ 2 /                                      
|--------------               otherwise              
|       2/5*a\                                       
|1 + cot |---|                                       
\        \ 2 /
{1for im⁡(a)=0∧5a mod 2π=0cot⁡2(5a2)−1cot⁡2(5a2)+1otherwise\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 5 a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{5 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1} & \text{otherwise} \end{cases}
//                     1                       for And(im(a) = 0, a mod pi = 0)\ //                       1                         for And(im(a) = 0, 3*a mod 2*pi = 0)\   //                      0                        for And(im(a) = 0, 2*a mod pi = 0)\ //                      0                        for And(im(a) = 0, 3*a mod pi = 0)\
||                                                                             | ||                                                                                     |   ||                                                                                 | ||                                                                                 |
|
(−({0for im⁡(a)=0∧2a mod π=0{0for im⁡(a)=0∧2a mod π=0sin⁡(2a)otherwiseotherwise)({0for im⁡(a)=0∧3a mod π=0{0for im⁡(a)=0∧3a mod π=0sin⁡(3a)otherwiseotherwise))+(({1for im⁡(a)=0∧a mod π=0{1for im⁡(a)=0∧a mod π=0cos⁡(2a)otherwiseotherwise)({1for im⁡(a)=0∧3a mod 2π=0{1for im⁡(a)=0∧3a mod 2π=0cos⁡(3a)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} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\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) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)
//      1        for And(im(a) = 0, a mod pi = 0)\ //      1        for And(im(a) = 0, 3*a mod 2*pi = 0)\                                                                                                  
||                                               | ||                                                   |   //   0      for And(im(a) = 0, 2*a mod pi = 0)\ //   0      for And(im(a) = 0, 3*a mod pi = 0)\
||      1                                        | ||      1                                            |   ||                                            | ||                                            |
|<-------------             otherwise            |*|<-------------               otherwise              | - |<   1                                        |*|<   1                                        |
||   /pi      \                                  | ||   /pi      \                                      |   ||--------              otherwise             | ||--------              otherwise             |
||csc|-- - 2*a|                                  | ||csc|-- - 3*a|                                      |   \\csc(2*a)                                    / \\csc(3*a)                                    /
\\   \2       /                                  / \\   \2       /                                      /
(−({0for im⁡(a)=0∧2a mod π=01csc⁡(2a)otherwise)({0for im⁡(a)=0∧3a mod π=01csc⁡(3a)otherwise))+(({1for im⁡(a)=0∧a mod π=01csc⁡(−2a+π2)otherwise)({1for im⁡(a)=0∧3a mod 2π=01csc⁡(−3a+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} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\frac{1}{\csc{\left(3 a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\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) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- 3 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)
        1                   1        
----------------- - -----------------
sec(2*a)*sec(3*a)   csc(2*a)*csc(3*a)
1sec⁡(2a)sec⁡(3a)−1csc⁡(2a)csc⁡(3a)\frac{1}{\sec{\left(2 a \right)} \sec{\left(3 a \right)}} - \frac{1}{\csc{\left(2 a \right)} \csc{\left(3 a \right)}}
cos(5*a)
cos⁡(5a)\cos{\left(5 a \right)}
/   1      for And(im(a) = 0, 5*a mod 2*pi = 0)
<                                              
\cos(5*a)               otherwise
{1for im⁡(a)=0∧5a mod 2π=0cos⁡(5a)otherwise\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 5 a \bmod 2 \pi = 0 \\\cos{\left(5 a \right)} & \text{otherwise} \end{cases}
             1                        1        
--------------------------- - -----------------
   /pi      \    /pi      \   csc(2*a)*csc(3*a)
csc|-- - 3*a|*csc|-- - 2*a|                    
   \2       /    \2       /
1csc⁡(−3a+π2)csc⁡(−2a+π2)−1csc⁡(2a)csc⁡(3a)\frac{1}{\csc{\left(- 3 a + \frac{\pi}{2} \right)} \csc{\left(- 2 a + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(2 a \right)} \csc{\left(3 a \right)}}
//      1        for And(im(a) = 0, a mod pi = 0)\ //      1        for And(im(a) = 0, 3*a mod 2*pi = 0)\                                                                                                  
||                                               | ||                                                   |   //   0      for And(im(a) = 0, 2*a mod pi = 0)\ //   0      for And(im(a) = 0, 3*a mod pi = 0)\
|<   /pi      \                                  |*|<   /pi      \                                      | - |<                                            |*|<                                            |
||sin|-- + 2*a|             otherwise            | ||sin|-- + 3*a|               otherwise              |   \\sin(2*a)              otherwise             / \\sin(3*a)              otherwise             /
\\   \2       /                                  / \\   \2       /                                      /
(−({0for im⁡(a)=0∧2a mod π=0sin⁡(2a)otherwise)({0for im⁡(a)=0∧3a mod π=0sin⁡(3a)otherwise))+(({1for im⁡(a)=0∧a mod π=0sin⁡(2a+π2)otherwise)({1for im⁡(a)=0∧3a mod 2π=0sin⁡(3a+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} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\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) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod 2 \pi = 0 \\\sin{\left(3 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)
   /pi      \    /pi      \                    
sin|-- + 2*a|*sin|-- + 3*a| - sin(2*a)*sin(3*a)
   \2       /    \2       /
−sin⁡(2a)sin⁡(3a)+sin⁡(2a+π2)sin⁡(3a+π2)- \sin{\left(2 a \right)} \sin{\left(3 a \right)} + \sin{\left(2 a + \frac{\pi}{2} \right)} \sin{\left(3 a + \frac{\pi}{2} \right)}
                       /      pi\    /      pi\
cos(2*a)*cos(3*a) - cos|2*a - --|*cos|3*a - --|
                       \      2 /    \      2 /
cos⁡(2a)cos⁡(3a)−cos⁡(2a−π2)cos⁡(3a−π2)\cos{\left(2 a \right)} \cos{\left(3 a \right)} - \cos{\left(2 a - \frac{\pi}{2} \right)} \cos{\left(3 a - \frac{\pi}{2} \right)}
                                                  //      1         for And(im(a) = 0, 3*a mod 2*pi = 0)\                                                      //      0        for And(im(a) = 0, 3*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)\ ||                                                 |
||                                              | ||        2/3*a\                                      |   ||                                               | ||       /3*a\                                     |
||        2                                     | ||-1 + cot |---|                                      |   ||  2*cot(a)                                     | ||  2*cot|---|                                     |
|<-1 + cot (a)                                  |*|<         \ 2 /                                      | - |<-----------              otherwise             |*|<       \ 2 /                                     |
||------------             otherwise            | ||--------------               otherwise              |   ||       2                                       | ||-------------              otherwise             |
||       2                                      | ||       2/3*a\                                       |   ||1 + cot (a)                                    | ||       2/3*a\                                    |
\\1 + cot (a)                                   / ||1 + cot |---|                                       |   \\                                               / ||1 + cot |---|                                    |
                                                  \\        \ 2 /                                       /                                                      \\        \ 2 /                                    /
(−({0for im⁡(a)=0∧2a mod π=02cot⁡(a)cot⁡2(a)+1otherwise)({0for im⁡(a)=0∧3a mod π=02cot⁡(3a2)cot⁡2(3a2)+1otherwise))+(({1for im⁡(a)=0∧a mod π=0cot⁡2(a)−1cot⁡2(a)+1otherwise)({1for im⁡(a)=0∧3a mod 2π=0cot⁡2(3a2)−1cot⁡2(3a2)+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} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\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) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)
   /pi      \
sin|-- + 5*a|
   \2       /
sin⁡(5a+π2)\sin{\left(5 a + \frac{\pi}{2} \right)}
                                                                                                  //      0        for And(im(a) = 0, 2*a mod pi = 0)\ //      0        for And(im(a) = 0, 3*a mod pi = 0)\
//   1      for And(im(a) = 0, a mod pi = 0)\ //   1      for And(im(a) = 0, 3*a mod 2*pi = 0)\   ||                                                 | ||                                                 |
||                                          | ||                                              |   ||      1                                          | ||      1                                          |
|<   1                                      |*|<   1                                          | - |<-------------              otherwise             |*|<-------------              otherwise             |
||--------             otherwise            | ||--------               otherwise              |   ||   /      pi\                                    | ||   /      pi\                                    |
\\sec(2*a)                                  / \\sec(3*a)                                      /   ||sec|2*a - --|                                    | ||sec|3*a - --|                                    |
                                                                                                  \\   \      2 /                                    / \\   \      2 /                                    /
(−({0for im⁡(a)=0∧2a mod π=01sec⁡(2a−π2)otherwise)({0for im⁡(a)=0∧3a mod π=01sec⁡(3a−π2)otherwise))+(({1for im⁡(a)=0∧a mod π=01sec⁡(2a)otherwise)({1for im⁡(a)=0∧3a mod 2π=01sec⁡(3a)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} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\frac{1}{\sec{\left(3 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\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) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(3 a \right)}} & \text{otherwise} \end{cases}\right)\right)
        1                        1             
----------------- - ---------------------------
sec(2*a)*sec(3*a)      /      pi\    /      pi\
                    sec|2*a - --|*sec|3*a - --|
                       \      2 /    \      2 /
−1sec⁡(2a−π2)sec⁡(3a−π2)+1sec⁡(2a)sec⁡(3a)- \frac{1}{\sec{\left(2 a - \frac{\pi}{2} \right)} \sec{\left(3 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 a \right)} \sec{\left(3 a \right)}}
Раскрыть выражение [src]
        3                      5           2                  4          
- 10*cos (a) + 3*cos(a) + 8*cos (a) - 6*sin (a)*cos(a) + 8*sin (a)*cos(a)
8sin⁡4(a)cos⁡(a)−6sin⁡2(a)cos⁡(a)+8cos⁡5(a)−10cos⁡3(a)+3cos⁡(a)8 \sin^{4}{\left(a \right)} \cos{\left(a \right)} - 6 \sin^{2}{\left(a \right)} \cos{\left(a \right)} + 8 \cos^{5}{\left(a \right)} - 10 \cos^{3}{\left(a \right)} + 3 \cos{\left(a \right)}