Дано

$$frac{sin{left (4 x right )} cos{left (4 x right )}}{sin{left (8 x right )}} + frac{cos^{2}{left (4 x right )} cos{left (8 x right )}}{sin^{2}{left (8 x right )}}$$
Подстановка условия
$$frac{sin{left (4 x right )} cos{left (4 x right )}}{sin{left (8 x right )}} + frac{cos^{2}{left (4 x right )} cos{left (8 x right )}}{sin^{2}{left (8 x right )}}$$

(cos(4*(1))^2*cos(8*(1)))/sin(8*(1))^2 + (cos(4*(1))*sin(4*(1)))/sin(8*(1))

$$frac{sin{left (4 (1) right )} cos{left (4 (1) right )}}{sin{left (8 (1) right )}} + frac{cos^{2}{left (4 (1) right )} cos{left (8 (1) right )}}{sin^{2}{left (8 (1) right )}}$$

(cos(4)^2*cos(8))/sin(8)^2 + (cos(4)*sin(4))/sin(8)

$$frac{cos^{2}{left (4 right )} cos{left (8 right )}}{sin^{2}{left (8 right )}} + frac{sin{left (4 right )} cos{left (4 right )}}{sin{left (8 right )}}$$

cos(4)^2*cos(8)/sin(8)^2 + cos(4)*sin(4)/sin(8)

$$frac{cos^{2}{left (4 right )} cos{left (8 right )}}{sin^{2}{left (8 right )}} + frac{sin{left (4 right )} cos{left (4 right )}}{sin{left (8 right )}}$$
Численный ответ

cos(4*x)^2*cos(8*x)/sin(8*x)^2 + cos(4*x)*sin(4*x)/sin(8*x)

Рациональный знаменатель
$$frac{1}{sin^{3}{left (8 x right )}} left(sin{left (4 x right )} sin^{2}{left (8 x right )} cos{left (4 x right )} + sin{left (8 x right )} cos^{2}{left (4 x right )} cos{left (8 x right )}right)$$
Объединение рациональных выражений
$$frac{cos{left (4 x right )}}{sin^{2}{left (8 x right )}} left(sin{left (4 x right )} sin{left (8 x right )} + cos{left (4 x right )} cos{left (8 x right )}right)$$
Общее упрощение

1
————————————
2
/ 2 2 2
64* -1 + 2*sin (x)/ *cos (x)*sin (x)

$$frac{1}{64 left(2 sin^{2}{left (x right )} – 1right)^{2} sin^{2}{left (x right )} cos^{2}{left (x right )}}$$
Соберем выражение
$$left(frac{1}{2} cos{left (8 x right )} + frac{1}{2}right) cot{left (8 x right )} csc{left (8 x right )} + frac{1}{2}$$
Общий знаменатель

2
cos (4*x)*cos(8*x) + cos(4*x)*sin(4*x)*sin(8*x)
———————————————–
2
sin (8*x)

$$frac{1}{sin^{2}{left (8 x right )}} left(sin{left (4 x right )} sin{left (8 x right )} cos{left (4 x right )} + cos^{2}{left (4 x right )} cos{left (8 x right )}right)$$
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Тригонометрическая часть

2
1 cos (4*x)*cos(8*x)
– + ——————
2 2
sin (8*x)

$$frac{1}{2} + frac{cos^{2}{left (4 x right )} cos{left (8 x right )}}{sin^{2}{left (8 x right )}}$$
Комбинаторика

(cos(4*x)*cos(8*x) + sin(4*x)*sin(8*x))*cos(4*x)
————————————————
2
sin (8*x)

$$frac{cos{left (4 x right )}}{sin^{2}{left (8 x right )}} left(sin{left (4 x right )} sin{left (8 x right )} + cos{left (4 x right )} cos{left (8 x right )}right)$$
Раскрыть выражение

2
/ 4 4 2 2 / 8 8 2 6 6 2 4 4 / 3 3 / 4 4 2 2
cos (x) + sin (x) – 6*cos (x)*sin (x)/ *cos (x) + sin (x) – 28*cos (x)*sin (x) – 28*cos (x)*sin (x) + 70*cos (x)*sin (x)/ – 4*sin (x)*cos(x) + 4*cos (x)*sin(x)/*cos (x) + sin (x) – 6*cos (x)*sin (x)/
————————————————————————————————————————— + ——————————————————————————-
2 5 3 7 7 3 5
/ 5 3 7 7 3 5 – 56*cos (x)*sin (x) – 8*sin (x)*cos(x) + 8*cos (x)*sin(x) + 56*cos (x)*sin (x)
– 56*cos (x)*sin (x) – 8*sin (x)*cos(x) + 8*cos (x)*sin(x) + 56*cos (x)*sin (x)/

$$frac{left(- 4 sin^{3}{left (x right )} cos{left (x right )} + 4 sin{left (x right )} cos^{3}{left (x right )}right) left(sin^{4}{left (x right )} – 6 sin^{2}{left (x right )} cos^{2}{left (x right )} + cos^{4}{left (x right )}right)}{- 8 sin^{7}{left (x right )} cos{left (x right )} + 56 sin^{5}{left (x right )} cos^{3}{left (x right )} – 56 sin^{3}{left (x right )} cos^{5}{left (x right )} + 8 sin{left (x right )} cos^{7}{left (x right )}} + frac{left(sin^{4}{left (x right )} – 6 sin^{2}{left (x right )} cos^{2}{left (x right )} + cos^{4}{left (x right )}right)^{2} left(sin^{8}{left (x right )} – 28 sin^{6}{left (x right )} cos^{2}{left (x right )} + 70 sin^{4}{left (x right )} cos^{4}{left (x right )} – 28 sin^{2}{left (x right )} cos^{6}{left (x right )} + cos^{8}{left (x right )}right)}{left(- 8 sin^{7}{left (x right )} cos{left (x right )} + 56 sin^{5}{left (x right )} cos^{3}{left (x right )} – 56 sin^{3}{left (x right )} cos^{5}{left (x right )} + 8 sin{left (x right )} cos^{7}{left (x right )}right)^{2}}$$
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4.81
Pomogashka
13 лет занимаюсь написанием курсовых, контрольных, дипломных работ, рефератов, отчетов по практике. Всегда доводила студентов до защиты. Оценки только положительные. Каждая работа уникальна и грамотно написана.Очень люблю свою работу.