sin((n+1)*x)/(2*(n+1))+si ... *x)/(2*(1-n)) если x=-1/3 (упростите выражение)

Вы ввели

sin((n + 1)*x)   sin((1 - n)*x)
-------------- + --------------
  2*(n + 1)        2*(1 - n)   

$$\frac{\sin{\left (x \left(n + 1\right) \right )}}{2 \left(n + 1\right)} + \frac{\sin{\left (x \left(- n + 1\right) \right )}}{2 \left(- n + 1\right)}$$

Подстановка условия

sin((n + 1)*x)/(2*(n + 1)) + sin((1 - n)*x)/(2*(1 - n)) при x = -1/3

sin((n + 1)*x)/(2*(n + 1)) + sin((1 - n)*x)/(2*(1 - n))

$$\frac{\sin{\left (x \left(n + 1\right) \right )}}{2 \left(n + 1\right)} + \frac{\sin{\left (x \left(- n + 1\right) \right )}}{2 \left(- n + 1\right)}$$

sin((n + 1)*(-1/3))/(2*(n + 1)) + sin((1 - n)*(-1/3))/(2*(1 - n))

$$\frac{\sin{\left ((-1/3) \left(n + 1\right) \right )}}{2 \left(n + 1\right)} + \frac{\sin{\left ((-1/3) \left(- n + 1\right) \right )}}{2 \left(- n + 1\right)}$$

sin((n + 1)*(-1)/3)/(2*(n + 1)) + sin((1 - n)*(-1)/3)/(2*(1 - n))

$$\frac{\sin{\left (\frac{1}{3} \left(- n - 1\right) \right )}}{2 \left(n + 1\right)} + \frac{\sin{\left (\frac{1}{3} \left(n - 1\right) \right )}}{2 \left(- n + 1\right)}$$

sin(-1/3 + n/3)/(2 - 2*n) - sin(1/3 + n/3)/(2 + 2*n)

$$- \frac{\sin{\left (\frac{n}{3} + \frac{1}{3} \right )}}{2 n + 2} + \frac{\sin{\left (\frac{n}{3} - \frac{1}{3} \right )}}{- 2 n + 2}$$

Степени

sin(x*(1 - n))   sin(x*(1 + n))
-------------- + --------------
   2 - 2*n          2 + 2*n    

$$\frac{\sin{\left (x \left(n + 1\right) \right )}}{2 n + 2} + \frac{\sin{\left (x \left(- n + 1\right) \right )}}{- 2 n + 2}$$

Численный ответ

sin((n + 1)*x)/(2.0 + 2.0*n) + sin((1 - n)*x)/(2.0 - 2.0*n)

Рациональный знаменатель

-2*sin(x + n*x) + 2*sin(-x + n*x) + 2*n*sin(x + n*x) + 2*n*sin(-x + n*x)
------------------------------------------------------------------------
                          (-2 + 2*n)*(2 + 2*n)                          

$$\frac{1}{\left(2 n - 2\right) \left(2 n + 2\right)} \left(2 n \sin{\left (n x - x \right )} + 2 n \sin{\left (n x + x \right )} + 2 \sin{\left (n x - x \right )} - 2 \sin{\left (n x + x \right )}\right)$$

Объединение рациональных выражений

(1 + n)*sin(x*(1 - n)) + (1 - n)*sin(x*(1 + n))
-----------------------------------------------
               2*(1 + n)*(1 - n)               

$$\frac{1}{2 \left(- n + 1\right) \left(n + 1\right)} \left(\left(- n + 1\right) \sin{\left (x \left(n + 1\right) \right )} + \left(n + 1\right) \sin{\left (x \left(- n + 1\right) \right )}\right)$$

Общее упрощение

(1 + n)*sin(x*(-1 + n)) + (-1 + n)*sin(x*(1 + n))
-------------------------------------------------
                2*(1 + n)*(-1 + n)               

$$\frac{1}{2 \left(n - 1\right) \left(n + 1\right)} \left(\left(n - 1\right) \sin{\left (x \left(n + 1\right) \right )} + \left(n + 1\right) \sin{\left (x \left(n - 1\right) \right )}\right)$$

Собрать выражение

sin(-x + n*x)   sin(x + n*x)
------------- + ------------
   -2 + 2*n       2 + 2*n   

$$\frac{\sin{\left (n x + x \right )}}{2 n + 2} + \frac{\sin{\left (n x - x \right )}}{2 n - 2}$$

Общий знаменатель

-sin(x + n*x) + n*sin(x + n*x) + n*sin(-x + n*x) + sin(-x + n*x)
----------------------------------------------------------------
                                   2                            
                           -2 + 2*n                             

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

Тригонометрическая часть

    1                          1                   
---------*sin((1 - n)*x) + ---------*sin((n + 1)*x)
2*(1 - n)                  2*(n + 1)               

$$\frac{1}{- 2 n + 2} \sin{\left (x \left(- n + 1\right) \right )} + \frac{1}{2 n + 2} \sin{\left (x \left(n + 1\right) \right )}$$

Комбинаторика

-sin(x + n*x) + n*sin(x + n*x) + n*sin(-x + n*x) + sin(-x + n*x)
----------------------------------------------------------------
                       2*(1 + n)*(-1 + n)                       

$$\frac{1}{2 \left(n - 1\right) \left(n + 1\right)} \left(n \sin{\left (n x - x \right )} + n \sin{\left (n x + x \right )} + \sin{\left (n x - x \right )} - \sin{\left (n x + x \right )}\right)$$

Раскрыть выражение

sin((1 - n)*x)   sin((n + 1)*x)
-------------- + --------------
  2*(1 - n)        2*(n + 1)   

$$\frac{\sin{\left (x \left(n + 1\right) \right )}}{2 n + 2} + \frac{\sin{\left (x \left(- n + 1\right) \right )}}{- 2 n + 2}$$

sin(x*(1 - n))   sin(x*(1 + n))
-------------- + --------------
   2 - 2*n          2 + 2*n    

$$\frac{\sin{\left (x \left(n + 1\right) \right )}}{2 n + 2} + \frac{\sin{\left (x \left(- n + 1\right) \right )}}{- 2 n + 2}$$