Дано

$$frac{- cos{left (2 a right )} + 1 – cos{left (6 a right )} + cos{left (8 a right )}}{- cos{left (a right )} cos{left (3 a right )} + cos{left (4 a right )}}$$
Подстановка условия
$$frac{- cos{left (2 a right )} + 1 – cos{left (6 a right )} + cos{left (8 a right )}}{- cos{left (a right )} cos{left (3 a right )} + cos{left (4 a right )}}$$

(1 – cos(2*(1/3)) – cos(6*(1/3)) + cos(8*(1/3)))/(cos(4*(1/3)) – cos((1/3))*cos(3*(1/3)))

$$frac{- cos{left (2 (1/3) right )} + 1 – cos{left (6 (1/3) right )} + cos{left (8 (1/3) right )}}{- cos{left ((1/3) right )} cos{left (3 (1/3) right )} + cos{left (4 (1/3) right )}}$$

(1 – cos(2/3) – cos(6/3) + cos(8/3))/(cos(4/3) – cos(1/3)*cos(3/3))

$$frac{cos{left (frac{8}{3} right )} + – cos{left (frac{2}{3} right )} + 1 – cos{left (frac{6}{3} right )}}{- cos{left (frac{1}{3} right )} cos{left (frac{3}{3} right )} + cos{left (frac{4}{3} right )}}$$

(1 – cos(2) – cos(2/3) + cos(8/3))/(-cos(1)*cos(1/3) + cos(4/3))

$$frac{cos{left (frac{8}{3} right )} – cos{left (frac{2}{3} right )} – cos{left (2 right )} + 1}{- cos{left (frac{1}{3} right )} cos{left (1 right )} + cos{left (frac{4}{3} right )}}$$
Степени
$$frac{- cos{left (2 a right )} – cos{left (6 a right )} + cos{left (8 a right )} + 1}{- cos{left (a right )} cos{left (3 a right )} + cos{left (4 a right )}}$$
Численный ответ

(1.0 – cos(2*a) – cos(6*a) + cos(8*a))/(-cos(a)*cos(3*a) + cos(4*a))

Рациональный знаменатель
$$frac{- cos{left (2 a right )} – cos{left (6 a right )} + cos{left (8 a right )} + 1}{- cos{left (a right )} cos{left (3 a right )} + cos{left (4 a right )}}$$
Объединение рациональных выражений
$$frac{- cos{left (2 a right )} – cos{left (6 a right )} + cos{left (8 a right )} + 1}{- cos{left (a right )} cos{left (3 a right )} + cos{left (4 a right )}}$$
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Общее упрощение

-1 – cos(8*a) + cos(2*a) + cos(6*a)
———————————–
-cos(4*a) + cos(a)*cos(3*a)

$$frac{cos{left (2 a right )} + cos{left (6 a right )} – cos{left (8 a right )} – 1}{cos{left (a right )} cos{left (3 a right )} – cos{left (4 a right )}}$$
Соберем выражение
$$- frac{2 cos{left (2 a right )}}{- cos{left (2 a right )} + cos{left (4 a right )}} – frac{2 cos{left (6 a right )}}{- cos{left (2 a right )} + cos{left (4 a right )}} + frac{2 cos{left (8 a right )}}{- cos{left (2 a right )} + cos{left (4 a right )}} + frac{2}{- cos{left (2 a right )} + cos{left (4 a right )}}$$

1 – cos(2*a) – cos(6*a) + cos(8*a)
———————————-
-cos(a)*cos(3*a) + cos(4*a)

$$frac{- cos{left (2 a right )} – cos{left (6 a right )} + cos{left (8 a right )} + 1}{- cos{left (a right )} cos{left (3 a right )} + cos{left (4 a right )}}$$
Общий знаменатель

-1 – cos(8*a) + cos(2*a) + cos(6*a)
———————————–
-cos(4*a) + cos(a)*cos(3*a)

$$frac{cos{left (2 a right )} + cos{left (6 a right )} – cos{left (8 a right )} – 1}{cos{left (a right )} cos{left (3 a right )} – cos{left (4 a right )}}$$
Тригонометрическая часть

2
-cos(6*a) + 2*sin (a) + cos(8*a)
——————————–
-cos(a)*cos(3*a) + cos(4*a)

$$frac{2 sin^{2}{left (a right )} – cos{left (6 a right )} + cos{left (8 a right )}}{- cos{left (a right )} cos{left (3 a right )} + cos{left (4 a right )}}$$
Комбинаторика

-1 – cos(8*a) + cos(2*a) + cos(6*a)
———————————–
-cos(4*a) + cos(a)*cos(3*a)

$$frac{cos{left (2 a right )} + cos{left (6 a right )} – cos{left (8 a right )} – 1}{cos{left (a right )} cos{left (3 a right )} – cos{left (4 a right )}}$$
Раскрыть выражение

8 2 6 8 2 6 2 6 6 2 2 4 4 2 4 4
1 + cos (a) + sin (a) + sin (a) + sin (a) – cos (a) – cos (a) – 28*cos (a)*sin (a) – 28*cos (a)*sin (a) – 15*cos (a)*sin (a) + 15*cos (a)*sin (a) + 70*cos (a)*sin (a)
———————————————————————————————————————————————————————-
4 4 / 3 2 2 2
cos (a) + sin (a) – cos (a) – 3*sin (a)*cos(a)/*cos(a) – 6*cos (a)*sin (a)

$$frac{1}{- left(- 3 sin^{2}{left (a right )} cos{left (a right )} + cos^{3}{left (a right )}right) cos{left (a right )} + sin^{4}{left (a right )} – 6 sin^{2}{left (a right )} cos^{2}{left (a right )} + cos^{4}{left (a right )}} left(sin^{8}{left (a right )} – 28 sin^{6}{left (a right )} cos^{2}{left (a right )} + sin^{6}{left (a right )} + 70 sin^{4}{left (a right )} cos^{4}{left (a right )} – 15 sin^{4}{left (a right )} cos^{2}{left (a right )} – 28 sin^{2}{left (a right )} cos^{6}{left (a right )} + 15 sin^{2}{left (a right )} cos^{4}{left (a right )} + sin^{2}{left (a right )} + cos^{8}{left (a right )} – cos^{6}{left (a right )} – cos^{2}{left (a right )} + 1right)$$
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