Общий знаменатель (1-sin(2*x-pi/3))/((1+sqr ... os(x)-(sqrt(3)-1)*sin(x))

Вы ввели

                  /      pi\           
           1 - sin|2*x - --|           
                  \      3 /           
---------------------------------------
/      ___\          /  ___    \       
\1 + \/ 3 /*cos(x) - \\/ 3  - 1/*sin(x)

$$\frac{- \sin{\left (2 x - \frac{\pi}{3} \right )} + 1}{- \left(-1 + \sqrt{3}\right) \sin{\left (x \right )} + \left(1 + \sqrt{3}\right) \cos{\left (x \right )}}$$

Степени

                  /      pi\            
           1 + cos|2*x + --|            
                  \      6 /            
----------------------------------------
/      ___\          /       ___\       
\1 + \/ 3 /*cos(x) - \-1 + \/ 3 /*sin(x)

$$\frac{\cos{\left (2 x + \frac{\pi}{6} \right )} + 1}{- \left(-1 + \sqrt{3}\right) \sin{\left (x \right )} + \left(1 + \sqrt{3}\right) \cos{\left (x \right )}}$$

Численный ответ

(1.0 - sin(2*x - pi/3))/(2.73205080756888*cos(x) - 0.732050807568877*sin(x))

Рациональный знаменатель

                     ___            ___                    /      pi\      /      pi\            ___           /      pi\     ___    /      pi\       
-cos(x) - sin(x) + \/ 3 *cos(x) - \/ 3 *sin(x) - cos(x)*cos|2*x + --| - cos|2*x + --|*sin(x) + \/ 3 *cos(x)*cos|2*x + --| - \/ 3 *cos|2*x + --|*sin(x)
                                                           \      6 /      \      6 /                          \      6 /            \      6 /       
------------------------------------------------------------------------------------------------------------------------------------------------------
                                                            2           2                                                                             
                                                       2*cos (x) + 2*sin (x) - 8*cos(x)*sin(x)                                                        

$$\frac{1}{2 \sin^{2}{\left (x \right )} - 8 \sin{\left (x \right )} \cos{\left (x \right )} + 2 \cos^{2}{\left (x \right )}} \left(- \sqrt{3} \sin{\left (x \right )} \cos{\left (2 x + \frac{\pi}{6} \right )} - \sin{\left (x \right )} \cos{\left (2 x + \frac{\pi}{6} \right )} - \sqrt{3} \sin{\left (x \right )} - \sin{\left (x \right )} - \cos{\left (x \right )} \cos{\left (2 x + \frac{\pi}{6} \right )} + \sqrt{3} \cos{\left (x \right )} \cos{\left (2 x + \frac{\pi}{6} \right )} - \cos{\left (x \right )} + \sqrt{3} \cos{\left (x \right )}\right)$$

Объединение рациональных выражений

                  /-pi + 6*x\           
           1 - sin|---------|           
                  \    3    /           
----------------------------------------
/      ___\          /       ___\       
\1 + \/ 3 /*cos(x) - \-1 + \/ 3 /*sin(x)

$$\frac{- \sin{\left (\frac{1}{3} \left(6 x - \pi\right) \right )} + 1}{- \left(-1 + \sqrt{3}\right) \sin{\left (x \right )} + \left(1 + \sqrt{3}\right) \cos{\left (x \right )}}$$

Общее упрощение

  ___ /       /      pi\\
\/ 2 *|1 + cos|2*x + --||
      \       \      6 //
-------------------------
           /    pi\      
      4*cos|x + --|      
           \    12/      

$$\frac{\sqrt{2} \left(\cos{\left (2 x + \frac{\pi}{6} \right )} + 1\right)}{4 \cos{\left (x + \frac{\pi}{12} \right )}}$$

Собрать выражение

                                                                      /      pi\                 
                                                                   cos|2*x + --|                 
                        1                                             \      6 /                 
- ---------------------------------------------- - ----------------------------------------------
                       ___            ___                               ___            ___       
  -cos(x) - sin(x) + \/ 3 *sin(x) - \/ 3 *cos(x)   -cos(x) - sin(x) + \/ 3 *sin(x) - \/ 3 *cos(x)

$$- \frac{\cos{\left (2 x + \frac{\pi}{6} \right )}}{- \sin{\left (x \right )} + \sqrt{3} \sin{\left (x \right )} - \sqrt{3} \cos{\left (x \right )} - \cos{\left (x \right )}} - \frac{1}{- \sin{\left (x \right )} + \sqrt{3} \sin{\left (x \right )} - \sqrt{3} \cos{\left (x \right )} - \cos{\left (x \right )}}$$

Общий знаменатель

                     /      pi\              
              1 + cos|2*x + --|              
                     \      6 /              
---------------------------------------------
  ___            ___                         
\/ 3 *cos(x) - \/ 3 *sin(x) + cos(x) + sin(x)

$$\frac{\cos{\left (2 x + \frac{\pi}{6} \right )} + 1}{- \sqrt{3} \sin{\left (x \right )} + \sin{\left (x \right )} + \cos{\left (x \right )} + \sqrt{3} \cos{\left (x \right )}}$$

Тригонометрическая часть

  ___ /       /      pi\\
\/ 2 *|1 + cos|2*x + --||
      \       \      6 //
-------------------------
           /    pi\      
      4*cos|x + --|      
           \    12/      

$$\frac{\sqrt{2} \left(\cos{\left (2 x + \frac{\pi}{6} \right )} + 1\right)}{4 \cos{\left (x + \frac{\pi}{12} \right )}}$$

Комбинаторика

                     /      pi\              
              1 + cos|2*x + --|              
                     \      6 /              
---------------------------------------------
  ___            ___                         
\/ 3 *cos(x) - \/ 3 *sin(x) + cos(x) + sin(x)

$$\frac{\cos{\left (2 x + \frac{\pi}{6} \right )} + 1}{- \sqrt{3} \sin{\left (x \right )} + \sin{\left (x \right )} + \cos{\left (x \right )} + \sqrt{3} \cos{\left (x \right )}}$$

Раскрыть выражение

                      ___               
         sin(2*x)   \/ 3 *cos(2*x)      
     1 - -------- + --------------      
            2             2             
----------------------------------------
/      ___\          /       ___\       
\1 + \/ 3 /*cos(x) - \-1 + \/ 3 /*sin(x)

$$\frac{- \frac{1}{2} \sin{\left (2 x \right )} + \frac{\sqrt{3}}{2} \cos{\left (2 x \right )} + 1}{- \left(-1 + \sqrt{3}\right) \sin{\left (x \right )} + \left(1 + \sqrt{3}\right) \cos{\left (x \right )}}$$