Общий знаменатель (1-cos(2*x))*(cos(pi/4+2* ... )/(2*(sin(2*x))^2-sin(x))

Вы ввели

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

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

Численный ответ

(1.0 - cos(2*x))*cos(pi/4 + 2*x)/(-sin(x) + 2.0*sin(2*x)^2)

Рациональный знаменатель

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

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

Объединение рациональных выражений

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

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

Общее упрощение

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

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

Собрать выражение

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

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

Комбинаторика

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

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

Общий знаменатель

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

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

Тригонометрическая часть

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

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

Раскрыть выражение

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

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