Дано
$$frac{2 cos{left (4 x right )}}{cos{left (8 x right )}} sin{left (4 x right )} + frac{2 sin^{2}{left (4 x right )}}{cos^{2}{left (8 x right )}} sin{left (8 x right )}$$
Подстановка условия
$$frac{2 cos{left (4 x right )}}{cos{left (8 x right )}} sin{left (4 x right )} + frac{2 sin^{2}{left (4 x right )}}{cos^{2}{left (8 x right )}} sin{left (8 x right )}$$
((2*cos(4*(3/2)))*sin(4*(3/2)))/cos(8*(3/2)) + ((2*sin(4*(3/2))^2)*sin(8*(3/2)))/cos(8*(3/2))^2
$$frac{2 cos{left (4 (3/2) right )}}{cos{left (8 (3/2) right )}} sin{left (4 (3/2) right )} + frac{2 sin^{2}{left (4 (3/2) right )}}{cos^{2}{left (8 (3/2) right )}} sin{left (8 (3/2) right )}$$
((2*cos(4*3/2))*sin(4*3/2))/cos(8*3/2) + ((2*sin(4*3/2)^2)*sin(8*3/2))/cos(8*3/2)^2
$$frac{2 cos{left (frac{12}{2} right )}}{cos{left (frac{24}{2} right )}} sin{left (frac{12}{2} right )} + frac{2 sin^{2}{left (frac{12}{2} right )}}{cos^{2}{left (frac{24}{2} right )}} sin{left (frac{24}{2} right )}$$
2*cos(6)*sin(6)/cos(12) + 2*sin(6)^2*sin(12)/cos(12)^2
$$frac{2 cos{left (6 right )}}{cos{left (12 right )}} sin{left (6 right )} + frac{2 sin^{2}{left (6 right )}}{cos^{2}{left (12 right )}} sin{left (12 right )}$$
Степени
$$frac{2 sin^{2}{left (4 x right )}}{cos^{2}{left (8 x right )}} sin{left (8 x right )} + frac{2 cos{left (4 x right )}}{cos{left (8 x right )}} sin{left (4 x right )}$$
Численный ответ
2.0*cos(4*x)*sin(4*x)/cos(8*x) + 2.0*sin(4*x)^2*sin(8*x)/cos(8*x)^2
Рациональный знаменатель
$$frac{1}{cos^{3}{left (8 x right )}} left(2 sin^{2}{left (4 x right )} sin{left (8 x right )} cos{left (8 x right )} + 2 sin{left (4 x right )} cos{left (4 x right )} cos^{2}{left (8 x right )}right)$$
Объединение рациональных выражений
$$frac{2 sin{left (4 x right )}}{cos^{2}{left (8 x right )}} left(sin{left (4 x right )} sin{left (8 x right )} + cos{left (4 x right )} cos{left (8 x right )}right)$$
Общее упрощение
/ 2 4
2*1 – 8*sin (x) + 8*sin (x)/*sin(4*x)
————————————–
2
cos (8*x)
$$frac{2 sin{left (4 x right )}}{cos^{2}{left (8 x right )}} left(8 sin^{4}{left (x right )} – 8 sin^{2}{left (x right )} + 1right)$$
Соберем выражение
$$left(- cos{left (8 x right )} + 1right) tan{left (8 x right )} sec{left (8 x right )} + tan{left (8 x right )}$$
Общий знаменатель
2
2*sin (4*x)*sin(8*x) + 2*cos(4*x)*cos(8*x)*sin(4*x)
—————————————————
2
cos (8*x)
$$frac{1}{cos^{2}{left (8 x right )}} left(2 sin^{2}{left (4 x right )} sin{left (8 x right )} + 2 sin{left (4 x right )} cos{left (4 x right )} cos{left (8 x right )}right)$$
Тригонометрическая часть
2
2*cos(4*x)*sin(4*x) 2*sin (4*x)*sin(8*x)
——————- + ——————–
cos(8*x) 2
cos (8*x)
$$frac{2 sin^{2}{left (4 x right )}}{cos^{2}{left (8 x right )}} sin{left (8 x right )} + frac{2 cos{left (4 x right )}}{cos{left (8 x right )}} sin{left (4 x right )}$$
Комбинаторика
2*(cos(4*x)*cos(8*x) + sin(4*x)*sin(8*x))*sin(4*x)
————————————————–
2
cos (8*x)
$$frac{2 sin{left (4 x right )}}{cos^{2}{left (8 x right )}} left(sin{left (4 x right )} sin{left (8 x right )} + cos{left (4 x right )} cos{left (8 x right )}right)$$
Раскрыть выражение
2
/ 3 3 / 4 4 2 2 / 3 3 / 5 3 7 7 3 5
– 8*sin (x)*cos(x) + 8*cos (x)*sin(x)/*cos (x) + sin (x) – 6*cos (x)*sin (x)/ 2* – 4*sin (x)*cos(x) + 4*cos (x)*sin(x)/ * – 56*cos (x)*sin (x) – 8*sin (x)*cos(x) + 8*cos (x)*sin(x) + 56*cos (x)*sin (x)/
——————————————————————————– + —————————————————————————————————————————-
8 8 2 6 6 2 4 4 2
cos (x) + sin (x) – 28*cos (x)*sin (x) – 28*cos (x)*sin (x) + 70*cos (x)*sin (x) / 8 8 2 6 6 2 4 4
cos (x) + sin (x) – 28*cos (x)*sin (x) – 28*cos (x)*sin (x) + 70*cos (x)*sin (x)/
$$frac{left(- 8 sin^{3}{left (x right )} cos{left (x right )} + 8 sin{left (x right )} cos^{3}{left (x right )}right) left(sin^{4}{left (x right )} – 6 sin^{2}{left (x right )} cos^{2}{left (x right )} + cos^{4}{left (x right )}right)}{sin^{8}{left (x right )} – 28 sin^{6}{left (x right )} cos^{2}{left (x right )} + 70 sin^{4}{left (x right )} cos^{4}{left (x right )} – 28 sin^{2}{left (x right )} cos^{6}{left (x right )} + cos^{8}{left (x right )}} + frac{2 left(- 4 sin^{3}{left (x right )} cos{left (x right )} + 4 sin{left (x right )} cos^{3}{left (x right )}right)^{2} left(- 8 sin^{7}{left (x right )} cos{left (x right )} + 56 sin^{5}{left (x right )} cos^{3}{left (x right )} – 56 sin^{3}{left (x right )} cos^{5}{left (x right )} + 8 sin{left (x right )} cos^{7}{left (x right )}right)}{left(sin^{8}{left (x right )} – 28 sin^{6}{left (x right )} cos^{2}{left (x right )} + 70 sin^{4}{left (x right )} cos^{4}{left (x right )} – 28 sin^{2}{left (x right )} cos^{6}{left (x right )} + cos^{8}{left (x right )}right)^{2}}$$
