Общий знаменатель -8*(2+15*sin(x)^6/cos(x)^ ... )^2+30*sin(x)^4/cos(x)^4)

Вы ввели

   /          6            2            4   \
   |    15*sin (x)   17*sin (x)   30*sin (x)|
-8*|2 + ---------- + ---------- + ----------|
   |        6            2            4     |
   \     cos (x)      cos (x)      cos (x)  /

$$- 8 \left(\frac{15 \sin^{6}{\left (x \right )}}{\cos^{6}{\left (x \right )}} + 2 + \frac{17 \sin^{2}{\left (x \right )}}{\cos^{2}{\left (x \right )}} + \frac{30 \sin^{4}{\left (x \right )}}{\cos^{4}{\left (x \right )}}\right)$$

Степени

             4             2             6   
      240*sin (x)   136*sin (x)   120*sin (x)
-16 - ----------- - ----------- - -----------
           4             2             6     
        cos (x)       cos (x)       cos (x)  

$$- \frac{120 \sin^{6}{\left (x \right )}}{\cos^{6}{\left (x \right )}} - \frac{240 \sin^{4}{\left (x \right )}}{\cos^{4}{\left (x \right )}} - \frac{136 \sin^{2}{\left (x \right )}}{\cos^{2}{\left (x \right )}} - 16$$

Численный ответ

-16.0 - 120.0*sin(x)^6/cos(x)^6 - 240.0*sin(x)^4/cos(x)^4 - 136.0*sin(x)^2/cos(x)^2

Рациональный знаменатель

         8       4           4    /   2    /     6            6   \         6       2   \
- 240*cos (x)*sin (x) - 8*cos (x)*\cos (x)*\2*cos (x) + 15*sin (x)/ + 17*cos (x)*sin (x)/
-----------------------------------------------------------------------------------------
                                            12                                           
                                         cos  (x)                                        

$$\frac{1}{\cos^{12}{\left (x \right )}} \left(- 8 \left(\left(15 \sin^{6}{\left (x \right )} + 2 \cos^{6}{\left (x \right )}\right) \cos^{2}{\left (x \right )} + 17 \sin^{2}{\left (x \right )} \cos^{6}{\left (x \right )}\right) \cos^{4}{\left (x \right )} - 240 \sin^{4}{\left (x \right )} \cos^{8}{\left (x \right )}\right)$$

Объединение рациональных выражений

   /     6            6            4       2            2       4   \
-8*\2*cos (x) + 15*sin (x) + 17*cos (x)*sin (x) + 30*cos (x)*sin (x)/
---------------------------------------------------------------------
                                  6                                  
                               cos (x)                               

$$- \frac{1}{\cos^{6}{\left (x \right )}} \left(120 \sin^{6}{\left (x \right )} + 240 \sin^{4}{\left (x \right )} \cos^{2}{\left (x \right )} + 136 \sin^{2}{\left (x \right )} \cos^{4}{\left (x \right )} + 16 \cos^{6}{\left (x \right )}\right)$$

Общее упрощение

 /           4            2   \ 
-\16 + 16*sin (x) + 88*sin (x)/ 
--------------------------------
               6                
            cos (x)             

$$- \frac{1}{\cos^{6}{\left (x \right )}} \left(16 \sin^{4}{\left (x \right )} + 88 \sin^{2}{\left (x \right )} + 16\right)$$

Собрать выражение

             4             2             6   
-16 - 240*tan (x) - 136*tan (x) - 120*tan (x)

$$- 120 \tan^{6}{\left (x \right )} - 240 \tan^{4}{\left (x \right )} - 136 \tan^{2}{\left (x \right )} - 16$$

              6              2              4   
      8*15*sin (x)   8*17*sin (x)   8*30*sin (x)
-16 - ------------ - ------------ - ------------
           6              2              4      
        cos (x)        cos (x)        cos (x)   

$$- \frac{136 \sin^{2}{\left (x \right )}}{\cos^{2}{\left (x \right )}} - \frac{240 \sin^{4}{\left (x \right )}}{\cos^{4}{\left (x \right )}} - \frac{120 \sin^{6}{\left (x \right )}}{\cos^{6}{\left (x \right )}} - 16$$

Общий знаменатель

             6             4       2             2       4   
      120*sin (x) + 136*cos (x)*sin (x) + 240*cos (x)*sin (x)
-16 - -------------------------------------------------------
                                 6                           
                              cos (x)                        

$$- \frac{1}{\cos^{6}{\left (x \right )}} \left(120 \sin^{6}{\left (x \right )} + 240 \sin^{4}{\left (x \right )} \cos^{2}{\left (x \right )} + 136 \sin^{2}{\left (x \right )} \cos^{4}{\left (x \right )}\right) - 16$$

Тригонометрическая часть

             4             2             6   
-16 - 240*tan (x) - 136*tan (x) - 120*tan (x)

$$- 120 \tan^{6}{\left (x \right )} - 240 \tan^{4}{\left (x \right )} - 136 \tan^{2}{\left (x \right )} - 16$$

Комбинаторика

   /   2         2   \ /     4            4            2       2   \
-8*\cos (x) + sin (x)/*\2*cos (x) + 15*sin (x) + 15*cos (x)*sin (x)/
--------------------------------------------------------------------
                                 6                                  
                              cos (x)                               

$$- \frac{8}{\cos^{6}{\left (x \right )}} \left(\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}\right) \left(15 \sin^{4}{\left (x \right )} + 15 \sin^{2}{\left (x \right )} \cos^{2}{\left (x \right )} + 2 \cos^{4}{\left (x \right )}\right)$$

Раскрыть выражение

             4             2             6   
      240*sin (x)   136*sin (x)   120*sin (x)
-16 - ----------- - ----------- - -----------
           4             2             6     
        cos (x)       cos (x)       cos (x)  

$$- \frac{120 \sin^{6}{\left (x \right )}}{\cos^{6}{\left (x \right )}} - \frac{240 \sin^{4}{\left (x \right )}}{\cos^{4}{\left (x \right )}} - \frac{136 \sin^{2}{\left (x \right )}}{\cos^{2}{\left (x \right )}} - 16$$