Найдите общий знаменатель для дробей -log(((2*sin(x))/(1+cos(x))-2*sqrt(5))/(2*sqrt(5)+(2*sin(x))/(1+cos(x))))/sqrt(5) (минус логарифм от (((2 умножить на синус от (х)) делить на (1 плюс косинус от (х)) минус 2 умножить на квадратный корень из (5)) делить на (2 умножить на квадратный корень из (5) плюс (2 умножить на синус от (х)) делить на (1 плюс косинус от (х)))) делить на квадратный корень из (5)) - найти с решением [Есть ответ!]

Общий знаменатель -log(((2*sin(x))/(1+cos(x ... (x))/(1+cos(x))))/sqrt(5)

Учитель очень удивится увидев твоё верное решение😉

Решение

Вы ввели [src]
    / 2*sin(x)        ___\ 
    |---------- - 2*\/ 5 | 
    |1 + cos(x)          | 
-log|--------------------| 
    |    ___    2*sin(x) | 
    |2*\/ 5  + ----------| 
    \          1 + cos(x)/ 
---------------------------
             ___           
           \/ 5            
$$\frac{1}{\sqrt{5}} \left(-1 \log{\left (\frac{- 2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}}{2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}\right)$$
Степени [src]
          /      ___    2*sin(x) \ 
          |- 2*\/ 5  + ----------| 
   ___    |            1 + cos(x)| 
-\/ 5 *log|----------------------| 
          |     ___    2*sin(x)  | 
          | 2*\/ 5  + ---------- | 
          \           1 + cos(x) / 
-----------------------------------
                 5                 
$$- \frac{\sqrt{5}}{5} \log{\left (\frac{- 2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}}{2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}$$
Численный ответ [src]
-0.447213595499958*log(((2*sin(x))/(1 + cos(x)) - 2*sqrt(5))/(2*sqrt(5) + (2*sin(x))/(1 + cos(x))))
Рациональный знаменатель [src]
          / /                                /                     2            2          \           /          2      \ /                     2            2          \           /          2      \                                     \ \ 
          | |  /     2                    2\ |                  sin (x)      sin (x)*cos(x)|      2    |       sin (x)   | |                  sin (x)      sin (x)*cos(x)|       ___ |       sin (x)   | /     2                    2\       | | 
          |-|5*\- sin (x) + 5*(1 + cos(x)) /*|5 + 5*cos(x) - ------------- - --------------| + sin (x)*|5 - -------------|*|5 + 5*cos(x) - ------------- - --------------| - 2*\/ 5 *|5 - -------------|*\- sin (x) + 5*(1 + cos(x)) /*sin(x)| | 
          | |                                |                           2               2 |           |                2| |                           2               2 |           |                2|                                     | | 
   ___    | \                                \               (1 + cos(x))    (1 + cos(x))  /           \    (1 + cos(x)) / \               (1 + cos(x))    (1 + cos(x))  /           \    (1 + cos(x)) /                                     / | 
-\/ 5 *log|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| 
          |                                                                 /          2      \                               /                     2            2          \                                                                  | 
          |                                                                 |       sin (x)   | /     2                    2\ |                  sin (x)      sin (x)*cos(x)|                                                                  | 
          |                                                                 |5 - -------------|*\- sin (x) + 5*(1 + cos(x)) /*|5 + 5*cos(x) - ------------- - --------------|                                                                  | 
          |                                                                 |                2|                               |                           2               2 |                                                                  | 
          \                                                                 \    (1 + cos(x)) /                               \               (1 + cos(x))    (1 + cos(x))  /                                                                  / 
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                        5                                                                                                                        
$$- \frac{\sqrt{5}}{5} \log{\left (- \frac{1}{\left(5 - \frac{\sin^{2}{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{2}}\right) \left(5 \left(\cos{\left (x \right )} + 1\right)^{2} - \sin^{2}{\left (x \right )}\right) \left(5 \cos{\left (x \right )} + 5 - \frac{\sin^{2}{\left (x \right )} \cos{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{2}} - \frac{\sin^{2}{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{2}}\right)} \left(- 2 \sqrt{5} \left(5 - \frac{\sin^{2}{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{2}}\right) \left(5 \left(\cos{\left (x \right )} + 1\right)^{2} - \sin^{2}{\left (x \right )}\right) \sin{\left (x \right )} + \left(5 - \frac{\sin^{2}{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{2}}\right) \left(5 \cos{\left (x \right )} + 5 - \frac{\sin^{2}{\left (x \right )} \cos{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{2}} - \frac{\sin^{2}{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{2}}\right) \sin^{2}{\left (x \right )} + 5 \left(5 \left(\cos{\left (x \right )} + 1\right)^{2} - \sin^{2}{\left (x \right )}\right) \left(5 \cos{\left (x \right )} + 5 - \frac{\sin^{2}{\left (x \right )} \cos{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{2}} - \frac{\sin^{2}{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{2}}\right)\right) \right )}$$
Объединение рациональных выражений [src]
          /    ___                      \ 
   ___    |- \/ 5 *(1 + cos(x)) + sin(x)| 
-\/ 5 *log|-----------------------------| 
          |   ___                       | 
          \ \/ 5 *(1 + cos(x)) + sin(x) / 
------------------------------------------
                    5                     
$$- \frac{\sqrt{5}}{5} \log{\left (\frac{- \sqrt{5} \left(\cos{\left (x \right )} + 1\right) + \sin{\left (x \right )}}{\sqrt{5} \left(\cos{\left (x \right )} + 1\right) + \sin{\left (x \right )}} \right )}$$
Общее упрощение [src]
          / /            ___             \ \ 
   ___    |-\-sin(x) + \/ 5 *(1 + cos(x))/ | 
-\/ 5 *log|--------------------------------| 
          |    ___                         | 
          \  \/ 5 *(1 + cos(x)) + sin(x)   / 
---------------------------------------------
                      5                      
$$- \frac{\sqrt{5}}{5} \log{\left (- \frac{\sqrt{5} \left(\cos{\left (x \right )} + 1\right) - \sin{\left (x \right )}}{\sqrt{5} \left(\cos{\left (x \right )} + 1\right) + \sin{\left (x \right )}} \right )}$$
Собрать выражение [src]
          / 2*sin(x)        ___\ 
          |---------- - 2*\/ 5 | 
   ___    |1 + cos(x)          | 
-\/ 5 *log|--------------------| 
          |    ___    2*sin(x) | 
          |2*\/ 5  + ----------| 
          \          1 + cos(x)/ 
---------------------------------
                5                
$$- \frac{\sqrt{5}}{5} \log{\left (\frac{- 2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}}{2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}$$
         /   ___     ___                 \
  ___    | \/ 5  + \/ 5 *cos(x) + sin(x) |
\/ 5 *log|-------------------------------|
         |    ___     ___                |
         \- \/ 5  - \/ 5 *cos(x) + sin(x)/
------------------------------------------
                    5                     
$$\frac{\sqrt{5}}{5} \log{\left (\frac{\sin{\left (x \right )} + \sqrt{5} \cos{\left (x \right )} + \sqrt{5}}{\sin{\left (x \right )} - \sqrt{5} \cos{\left (x \right )} - \sqrt{5}} \right )}$$
          /                                                                                                                                                                  ___                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                ___                                                                                                         ___                                                                                                        ___                                                                                                          ___                                                                                                        ___                                                       \ 
   ___    |                                                   sin(3*x)                                                                                                  24*\/ 5                                                                                                10*sin(2*x)                                                                                                  3*sin(x)                                                                                                   25*sin(x)                                                                                                    5*sin(3*x)                                                                                             16*\/ 5 *cos(2*x)                                                                                           15*\/ 5 *cos(x)                                                                                             5*\/ 5 *cos(3*x)                                                                                            59*\/ 5 *cos(x)                                                                                               \/ 5 *cos(3*x)                                              | 
-\/ 5 *log|- --------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------ - ------------------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------ - --------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------| 
          |                     ___                                 ___                 ___                 ___                            ___                               ___                 ___                 ___                            ___                               ___                 ___                 ___                             ___                                 ___                 ___                 ___            /                  ___                               ___                 ___                 ___       \     /                  ___                               ___                 ___                 ___       \                     ___                               ___                 ___                 ___                             ___                                 ___                 ___                 ___                             ___                                 ___                 ___                 ___            /                  ___                               ___                 ___                 ___       \     /                  ___                               ___                 ___                 ___       \| 
          \  14*sin(3*x) + 56*\/ 5  + 60*sin(2*x) + 78*sin(x) + 2*\/ 5 *cos(3*x) + 24*\/ 5 *cos(2*x) + 78*\/ 5 *cos(x)   7*sin(3*x) + 28*\/ 5  + 30*sin(2*x) + 39*sin(x) + \/ 5 *cos(3*x) + 12*\/ 5 *cos(2*x) + 39*\/ 5 *cos(x)   7*sin(3*x) + 28*\/ 5  + 30*sin(2*x) + 39*sin(x) + \/ 5 *cos(3*x) + 12*\/ 5 *cos(2*x) + 39*\/ 5 *cos(x)   14*sin(3*x) + 56*\/ 5  + 60*sin(2*x) + 78*sin(x) + 2*\/ 5 *cos(3*x) + 24*\/ 5 *cos(2*x) + 78*\/ 5 *cos(x)   2*\7*sin(3*x) + 28*\/ 5  + 30*sin(2*x) + 39*sin(x) + \/ 5 *cos(3*x) + 12*\/ 5 *cos(2*x) + 39*\/ 5 *cos(x)/   2*\7*sin(3*x) + 28*\/ 5  + 30*sin(2*x) + 39*sin(x) + \/ 5 *cos(3*x) + 12*\/ 5 *cos(2*x) + 39*\/ 5 *cos(x)/   7*sin(3*x) + 28*\/ 5  + 30*sin(2*x) + 39*sin(x) + \/ 5 *cos(3*x) + 12*\/ 5 *cos(2*x) + 39*\/ 5 *cos(x)   14*sin(3*x) + 56*\/ 5  + 60*sin(2*x) + 78*sin(x) + 2*\/ 5 *cos(3*x) + 24*\/ 5 *cos(2*x) + 78*\/ 5 *cos(x)   14*sin(3*x) + 56*\/ 5  + 60*sin(2*x) + 78*sin(x) + 2*\/ 5 *cos(3*x) + 24*\/ 5 *cos(2*x) + 78*\/ 5 *cos(x)   2*\7*sin(3*x) + 28*\/ 5  + 30*sin(2*x) + 39*sin(x) + \/ 5 *cos(3*x) + 12*\/ 5 *cos(2*x) + 39*\/ 5 *cos(x)/   2*\7*sin(3*x) + 28*\/ 5  + 30*sin(2*x) + 39*sin(x) + \/ 5 *cos(3*x) + 12*\/ 5 *cos(2*x) + 39*\/ 5 *cos(x)// 
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     5                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
$$- \frac{\sqrt{5}}{5} \log{\left (\frac{3 \sin{\left (x \right )}}{78 \sin{\left (x \right )} + 60 \sin{\left (2 x \right )} + 14 \sin{\left (3 x \right )} + 78 \sqrt{5} \cos{\left (x \right )} + 24 \sqrt{5} \cos{\left (2 x \right )} + 2 \sqrt{5} \cos{\left (3 x \right )} + 56 \sqrt{5}} - \frac{\sin{\left (3 x \right )}}{78 \sin{\left (x \right )} + 60 \sin{\left (2 x \right )} + 14 \sin{\left (3 x \right )} + 78 \sqrt{5} \cos{\left (x \right )} + 24 \sqrt{5} \cos{\left (2 x \right )} + 2 \sqrt{5} \cos{\left (3 x \right )} + 56 \sqrt{5}} - \frac{15 \sqrt{5} \cos{\left (x \right )}}{78 \sin{\left (x \right )} + 60 \sin{\left (2 x \right )} + 14 \sin{\left (3 x \right )} + 78 \sqrt{5} \cos{\left (x \right )} + 24 \sqrt{5} \cos{\left (2 x \right )} + 2 \sqrt{5} \cos{\left (3 x \right )} + 56 \sqrt{5}} - \frac{5 \sqrt{5} \cos{\left (3 x \right )}}{78 \sin{\left (x \right )} + 60 \sin{\left (2 x \right )} + 14 \sin{\left (3 x \right )} + 78 \sqrt{5} \cos{\left (x \right )} + 24 \sqrt{5} \cos{\left (2 x \right )} + 2 \sqrt{5} \cos{\left (3 x \right )} + 56 \sqrt{5}} - \frac{25 \sin{\left (x \right )}}{78 \sin{\left (x \right )} + 60 \sin{\left (2 x \right )} + 14 \sin{\left (3 x \right )} + 78 \sqrt{5} \cos{\left (x \right )} + 24 \sqrt{5} \cos{\left (2 x \right )} + 2 \sqrt{5} \cos{\left (3 x \right )} + 56 \sqrt{5}} - \frac{10 \sin{\left (2 x \right )}}{39 \sin{\left (x \right )} + 30 \sin{\left (2 x \right )} + 7 \sin{\left (3 x \right )} + 39 \sqrt{5} \cos{\left (x \right )} + 12 \sqrt{5} \cos{\left (2 x \right )} + \sqrt{5} \cos{\left (3 x \right )} + 28 \sqrt{5}} - \frac{5 \sin{\left (3 x \right )}}{78 \sin{\left (x \right )} + 60 \sin{\left (2 x \right )} + 14 \sin{\left (3 x \right )} + 78 \sqrt{5} \cos{\left (x \right )} + 24 \sqrt{5} \cos{\left (2 x \right )} + 2 \sqrt{5} \cos{\left (3 x \right )} + 56 \sqrt{5}} - \frac{59 \sqrt{5} \cos{\left (x \right )}}{78 \sin{\left (x \right )} + 60 \sin{\left (2 x \right )} + 14 \sin{\left (3 x \right )} + 78 \sqrt{5} \cos{\left (x \right )} + 24 \sqrt{5} \cos{\left (2 x \right )} + 2 \sqrt{5} \cos{\left (3 x \right )} + 56 \sqrt{5}} - \frac{16 \sqrt{5} \cos{\left (2 x \right )}}{39 \sin{\left (x \right )} + 30 \sin{\left (2 x \right )} + 7 \sin{\left (3 x \right )} + 39 \sqrt{5} \cos{\left (x \right )} + 12 \sqrt{5} \cos{\left (2 x \right )} + \sqrt{5} \cos{\left (3 x \right )} + 28 \sqrt{5}} - \frac{\sqrt{5} \cos{\left (3 x \right )}}{78 \sin{\left (x \right )} + 60 \sin{\left (2 x \right )} + 14 \sin{\left (3 x \right )} + 78 \sqrt{5} \cos{\left (x \right )} + 24 \sqrt{5} \cos{\left (2 x \right )} + 2 \sqrt{5} \cos{\left (3 x \right )} + 56 \sqrt{5}} - \frac{24 \sqrt{5}}{39 \sin{\left (x \right )} + 30 \sin{\left (2 x \right )} + 7 \sin{\left (3 x \right )} + 39 \sqrt{5} \cos{\left (x \right )} + 12 \sqrt{5} \cos{\left (2 x \right )} + \sqrt{5} \cos{\left (3 x \right )} + 28 \sqrt{5}} \right )}$$
Комбинаторика [src]
          /            ___                                                                 \ 
   ___    |        2*\/ 5                                   2*sin(x)                       | 
-\/ 5 *log|- -------------------- + -------------------------------------------------------| 
          |      ___    2*sin(x)        ___       ___           2*sin(x)    2*cos(x)*sin(x)| 
          |  2*\/ 5  + ----------   2*\/ 5  + 2*\/ 5 *cos(x) + ---------- + ---------------| 
          \            1 + cos(x)                              1 + cos(x)      1 + cos(x)  / 
---------------------------------------------------------------------------------------------
                                              5                                              
$$- \frac{\sqrt{5}}{5} \log{\left (\frac{2 \sin{\left (x \right )}}{2 \sqrt{5} \cos{\left (x \right )} + 2 \sqrt{5} + \frac{2 \sin{\left (x \right )} \cos{\left (x \right )}}{\cos{\left (x \right )} + 1} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} - \frac{2 \sqrt{5}}{2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}$$
Общий знаменатель [src]
          /                                                            ___       \ 
   ___    |                      sin(x)                              \/ 5        | 
-\/ 5 *log|------------------------------------------------- - ------------------| 
          |  ___     ___            sin(x)     cos(x)*sin(x)     ___     sin(x)  | 
          |\/ 5  + \/ 5 *cos(x) + ---------- + -------------   \/ 5  + ----------| 
          \                       1 + cos(x)     1 + cos(x)            1 + cos(x)/ 
-----------------------------------------------------------------------------------
                                         5                                         
$$- \frac{\sqrt{5}}{5} \log{\left (\frac{\sin{\left (x \right )}}{\sqrt{5} \cos{\left (x \right )} + \sqrt{5} + \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{\cos{\left (x \right )} + 1} + \frac{\sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} - \frac{\sqrt{5}}{\sqrt{5} + \frac{\sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}$$
Тригонометрическая часть [src]
          /    ___     ___                \ 
   ___    |- \/ 5  - \/ 5 *cos(x) + sin(x)| 
-\/ 5 *log|-------------------------------| 
          |   ___     ___                 | 
          \ \/ 5  + \/ 5 *cos(x) + sin(x) / 
--------------------------------------------
                     5                      
$$- \frac{\sqrt{5}}{5} \log{\left (\frac{\sin{\left (x \right )} - \sqrt{5} \cos{\left (x \right )} - \sqrt{5}}{\sin{\left (x \right )} + \sqrt{5} \cos{\left (x \right )} + \sqrt{5}} \right )}$$
Раскрыть выражение [src]
          /      ___    2*sin(x) \ 
          |- 2*\/ 5  + ----------| 
   ___    |            1 + cos(x)| 
-\/ 5 *log|----------------------| 
          |     ___    2*sin(x)  | 
          | 2*\/ 5  + ---------- | 
          \           1 + cos(x) / 
-----------------------------------
                 5                 
$$- \frac{\sqrt{5}}{5} \log{\left (\frac{- 2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}}{2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}$$
  ___ /     /      ___    2*sin(x) \      /    ___    2*sin(x) \\
\/ 5 *|- log|- 2*\/ 5  + ----------| + log|2*\/ 5  + ----------||
      \     \            1 + cos(x)/      \          1 + cos(x)//
-----------------------------------------------------------------
                                5                                
$$\frac{\sqrt{5}}{5} \left(- \log{\left (- 2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1} \right )} + \log{\left (2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1} \right )}\right)$$