Общий знаменатель ((1-cos(2*x))*cos((45*pi/ ... )/(2*sin(2*x)^(2)-sin(x))

Вы ввели

                  /45*pi      \
(1 - cos(2*x))*cos|----- + 2*x|
                  \ 360       /
-------------------------------
           2                   
      2*sin (2*x) - sin(x)     

$$\frac{\left(- \cos{\left (2 x \right )} + 1\right) \cos{\left (2 x + \frac{45 \pi}{360} \right )}}{- \sin{\left (x \right )} + 2 \sin^{2}{\left (2 x \right )}}$$

Степени

                  /      pi\
(1 - cos(2*x))*cos|2*x + --|
                  \      8 /
----------------------------
                  2         
   -sin(x) + 2*sin (2*x)    

$$\frac{\left(- \cos{\left (2 x \right )} + 1\right) \cos{\left (2 x + \frac{\pi}{8} \right )}}{- \sin{\left (x \right )} + 2 \sin^{2}{\left (2 x \right )}}$$

Численный ответ

(1.0 - cos(2*x))*cos((45*pi)/360 + 2*x)/(-sin(x) + 2.0*sin(2*x)^2)

Рациональный знаменатель

                   /      pi\
(-1 + cos(2*x))*cos|2*x + --|
                   \      8 /
-----------------------------
           2                 
    - 2*sin (2*x) + sin(x)   

$$\frac{\left(\cos{\left (2 x \right )} - 1\right) \cos{\left (2 x + \frac{\pi}{8} \right )}}{\sin{\left (x \right )} - 2 \sin^{2}{\left (2 x \right )}}$$

Объединение рациональных выражений

                  /pi + 16*x\
(1 - cos(2*x))*cos|---------|
                  \    8    /
-----------------------------
                   2         
    -sin(x) + 2*sin (2*x)    

$$\frac{\left(- \cos{\left (2 x \right )} + 1\right) \cos{\left (\frac{1}{8} \left(16 x + \pi\right) \right )}}{- \sin{\left (x \right )} + 2 \sin^{2}{\left (2 x \right )}}$$

Общее упрощение

                   /      pi\
(-1 + cos(2*x))*cos|2*x + --|
                   \      8 /
-----------------------------
    -1 + cos(4*x) + sin(x)   

$$\frac{\left(\cos{\left (2 x \right )} - 1\right) \cos{\left (2 x + \frac{\pi}{8} \right )}}{\sin{\left (x \right )} + \cos{\left (4 x \right )} - 1}$$

Собрать выражение

          ___________                                                           
         /       ___                                                            
        /  1   \/ 2                   /      pi\                 /      pi\     
       /   - + -----               cos|4*x + --|              cos|2*x + --|     
     \/    2     4                    \      8 /                 \      8 /     
-------------------------- + -------------------------- - ----------------------
-2 + 2*cos(4*x) + 2*sin(x)   -2 + 2*cos(4*x) + 2*sin(x)   -1 + cos(4*x) + sin(x)

$$\frac{\cos{\left (4 x + \frac{\pi}{8} \right )}}{2 \sin{\left (x \right )} + 2 \cos{\left (4 x \right )} - 2} + \frac{\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2 \sin{\left (x \right )} + 2 \cos{\left (4 x \right )} - 2} - \frac{\cos{\left (2 x + \frac{\pi}{8} \right )}}{\sin{\left (x \right )} + \cos{\left (4 x \right )} - 1}$$

Общий знаменатель

 /     /      pi\               /      pi\\ 
-|- cos|2*x + --| + cos(2*x)*cos|2*x + --|| 
 \     \      8 /               \      8 // 
--------------------------------------------
                          2                 
           -sin(x) + 2*sin (2*x)            

$$- \frac{\cos{\left (2 x \right )} \cos{\left (2 x + \frac{\pi}{8} \right )} - \cos{\left (2 x + \frac{\pi}{8} \right )}}{- \sin{\left (x \right )} + 2 \sin^{2}{\left (2 x \right )}}$$

Тригонометрическая часть

     2       /45*pi      \
2*sin (x)*cos|----- + 2*x|
             \ 360       /
--------------------------
        2                 
   2*sin (2*x) - sin(x)   

$$\frac{2 \sin^{2}{\left (x \right )} \cos{\left (2 x + \frac{45 \pi}{360} \right )}}{- \sin{\left (x \right )} + 2 \sin^{2}{\left (2 x \right )}}$$

Комбинаторика

                    /      pi\ 
-(-1 + cos(2*x))*cos|2*x + --| 
                    \      8 / 
-------------------------------
                    2          
     -sin(x) + 2*sin (2*x)     

$$- \frac{\left(\cos{\left (2 x \right )} - 1\right) \cos{\left (2 x + \frac{\pi}{8} \right )}}{- \sin{\left (x \right )} + 2 \sin^{2}{\left (2 x \right )}}$$

Раскрыть выражение

/     ___________                 ___________         \                        
|    /       ___                 /       ___          |                        
|   /  1   \/ 2                 /  1   \/ 2           | /       2         2   \
|  /   - + ----- *cos(2*x) -   /   - - ----- *sin(2*x)|*\1 + sin (x) - cos (x)/
\\/    2     4               \/    2     4            /                        
-------------------------------------------------------------------------------
                                         2       2                             
                          -sin(x) + 8*cos (x)*sin (x)                          

$$\frac{1}{8 \sin^{2}{\left (x \right )} \cos^{2}{\left (x \right )} - \sin{\left (x \right )}} \left(- \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}} \sin{\left (2 x \right )} + \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \cos{\left (2 x \right )}\right) \left(\sin^{2}{\left (x \right )} - \cos^{2}{\left (x \right )} + 1\right)$$