Дано
$$frac{-1 cos{left (x left(n + 2right) right )}}{2 left(n + 2right)} – frac{1}{- 2 n + 4} cos{left (x left(- n + 2right) right )}$$
Подстановка условия
$$frac{-1 cos{left (x left(n + 2right) right )}}{2 left(n + 2right)} – frac{1}{- 2 n + 4} cos{left (x left(- n + 2right) right )}$$
(-cos((n + 2)*(-1/4)))/(2*(2 + n)) – cos((2 – n)*(-1/4))/2*(2 – n)
$$frac{-1 cos{left ((-1/4) left(n + 2right) right )}}{2 left(n + 2right)} – frac{1}{- 2 n + 4} cos{left ((-1/4) left(- n + 2right) right )}$$
(-cos((n + 2)*(-1)/4))/(2*(2 + n)) – cos((2 – n)*(-1)/4)/2*(2 – n)
$$frac{1}{2 left(n + 2right)} left(-1 cos{left (frac{1}{4} left(- n – 2right) right )}right) – frac{1}{- 2 n + 4} cos{left (frac{1}{4} left(n – 2right) right )}$$
-cos(-1/2 + n/4)/(4 – 2*n) – cos(1/2 + n/4)/(4 + 2*n)
$$- frac{cos{left (frac{n}{4} + frac{1}{2} right )}}{2 n + 4} – frac{cos{left (frac{n}{4} – frac{1}{2} right )}}{- 2 n + 4}$$
Степени
$$- frac{cos{left (x left(n + 2right) right )}}{2 n + 4} – frac{cos{left (x left(- n + 2right) right )}}{- 2 n + 4}$$
Численный ответ
-cos((2 – n)*x)/(4.0 – 2.0*n) – 1.0*cos((n + 2)*x)/(4.0 + 2.0*n)
Рациональный знаменатель
$$frac{1}{left(2 n – 4right) left(2 n + 4right)} left(2 n cos{left (n x – 2 x right )} – 2 n cos{left (n x + 2 x right )} + 4 cos{left (n x – 2 x right )} + 4 cos{left (n x + 2 x right )}right)$$
Объединение рациональных выражений
$$frac{1}{2 left(- n + 2right) left(n + 2right)} left(- left(- n + 2right) cos{left (x left(n + 2right) right )} – left(n + 2right) cos{left (x left(- n + 2right) right )}right)$$
Общее упрощение
(2 + n)*cos(x*(-2 + n)) + (2 – n)*cos(x*(2 + n))
————————————————
2*(-2 + n)*(2 + n)
$$frac{1}{2 left(n – 2right) left(n + 2right)} left(left(- n + 2right) cos{left (x left(n + 2right) right )} + left(n + 2right) cos{left (x left(n – 2right) right )}right)$$
Соберем выражение
$$frac{n}{2} cos{left (n x – 2 x right )} – cos{left (n x – 2 x right )} – frac{cos{left (n x + 2 x right )}}{2 n + 4}$$
Комбинаторика
-(-2*cos(-2*x + n*x) – 2*cos(2*x + n*x) + n*cos(2*x + n*x) – n*cos(-2*x + n*x))
——————————————————————————–
2*(-2 + n)*(2 + n)
$$- frac{1}{2 left(n – 2right) left(n + 2right)} left(- n cos{left (n x – 2 x right )} + n cos{left (n x + 2 x right )} – 2 cos{left (n x – 2 x right )} – 2 cos{left (n x + 2 x right )}right)$$
Общий знаменатель
-(-2*cos(-2*x + n*x) – 2*cos(2*x + n*x) + n*cos(2*x + n*x) – n*cos(-2*x + n*x))
——————————————————————————–
2
-8 + 2*n
$$- frac{1}{2 n^{2} – 8} left(- n cos{left (n x – 2 x right )} + n cos{left (n x + 2 x right )} – 2 cos{left (n x – 2 x right )} – 2 cos{left (n x + 2 x right )}right)$$
Тригонометрическая часть
1 cos(x*(-2 + n))
– ———*cos((n + 2)*x) – —————
2*(2 + n) 2*(2 – n)
$$- frac{cos{left (x left(n + 2right) right )}}{2 n + 4} – frac{cos{left (x left(n – 2right) right )}}{- 2 n + 4}$$
Раскрыть выражение
cos((n + 2)*x) cos((2 – n)*x)
– ————– – ————–
2*(2 + n) 2*(2 – n)
$$- frac{cos{left (x left(n + 2right) right )}}{2 n + 4} – frac{cos{left (x left(- n + 2right) right )}}{- 2 n + 4}$$
cos(x*(2 + n)) (2 – n)*cos(x*(2 – n))
– ————– – ———————-
4 + 2*n 2
$$- frac{1}{2} left(- n + 2right) cos{left (x left(- n + 2right) right )} – frac{cos{left (x left(n + 2right) right )}}{2 n + 4}$$
