Дано

$$sin^{2}{left (x right )} cos^{3}{left (x right )} left(20 sin^{4}{left (x right )} + 12 cos^{4}{left (x right )} – 49 sin^{2}{left (x right )} cos^{2}{left (x right )}right)$$
Подстановка условия
$$sin^{2}{left (x right )} cos^{3}{left (x right )} left(20 sin^{4}{left (x right )} + 12 cos^{4}{left (x right )} – 49 sin^{2}{left (x right )} cos^{2}{left (x right )}right)$$

(cos((1))^3*sin((1))^2)*(12*cos((1))^4 + 20*sin((1))^4 – 49*cos((1))^2*sin((1))^2)

$$sin^{2}{left ((1) right )} cos^{3}{left ((1) right )} left(20 sin^{4}{left ((1) right )} + 12 cos^{4}{left ((1) right )} – 49 sin^{2}{left ((1) right )} cos^{2}{left ((1) right )}right)$$

(cos(1)^3*sin(1)^2)*(12*cos(1)^4 + 20*sin(1)^4 – 49*cos(1)^2*sin(1)^2)

$$sin^{2}{left (1 right )} cos^{3}{left (1 right )} left(- 49 sin^{2}{left (1 right )} cos^{2}{left (1 right )} + 12 cos^{4}{left (1 right )} + 20 sin^{4}{left (1 right )}right)$$

cos(1)^3*sin(1)^2*(12*cos(1)^4 + 20*sin(1)^4 – 49*cos(1)^2*sin(1)^2)

$$left(- 49 sin^{2}{left (1 right )} cos^{2}{left (1 right )} + 12 cos^{4}{left (1 right )} + 20 sin^{4}{left (1 right )}right) sin^{2}{left (1 right )} cos^{3}{left (1 right )}$$
Степени
$$left(20 sin^{4}{left (x right )} – 49 sin^{2}{left (x right )} cos^{2}{left (x right )} + 12 cos^{4}{left (x right )}right) sin^{2}{left (x right )} cos^{3}{left (x right )}$$
Численный ответ

cos(x)^3*sin(x)^2*(12.0*cos(x)^4 + 20.0*sin(x)^4 – 49.0*cos(x)^2*sin(x)^2)

Рациональный знаменатель
$$left(20 sin^{4}{left (x right )} – 49 sin^{2}{left (x right )} cos^{2}{left (x right )} + 12 cos^{4}{left (x right )}right) sin^{2}{left (x right )} cos^{3}{left (x right )}$$
Объединение рациональных выражений
$$left(20 sin^{4}{left (x right )} – 49 sin^{2}{left (x right )} cos^{2}{left (x right )} + 12 cos^{4}{left (x right )}right) sin^{2}{left (x right )} cos^{3}{left (x right )}$$
Общее упрощение

3 2 / 2 4
cos (x)*sin (x)*12 – 73*sin (x) + 81*sin (x)/

$$left(81 sin^{4}{left (x right )} – 73 sin^{2}{left (x right )} + 12right) sin^{2}{left (x right )} cos^{3}{left (x right )}$$
Соберем выражение
$$- frac{3}{128} cos{left (x right )} + frac{9}{64} cos{left (3 x right )} + frac{25}{64} cos{left (5 x right )} – frac{49}{256} cos{left (7 x right )} – frac{81}{256} cos{left (9 x right )}$$

3 2 / 4 4 2 2
cos (x)*sin (x)*12*cos (x) + 20*sin (x) – 49*cos (x)*sin (x)/

$$left(20 sin^{4}{left (x right )} – 49 sin^{2}{left (x right )} cos^{2}{left (x right )} + 12 cos^{4}{left (x right )}right) sin^{2}{left (x right )} cos^{3}{left (x right )}$$
Общий знаменатель

5 4 7 2 3 6
– 49*cos (x)*sin (x) + 12*cos (x)*sin (x) + 20*cos (x)*sin (x)

$$20 sin^{6}{left (x right )} cos^{3}{left (x right )} – 49 sin^{4}{left (x right )} cos^{5}{left (x right )} + 12 sin^{2}{left (x right )} cos^{7}{left (x right )}$$
Тригонометрическая часть

3 2 / 2 4
cos (x)*sin (x)*12 – 73*sin (x) + 81*sin (x)/

$$left(81 sin^{4}{left (x right )} – 73 sin^{2}{left (x right )} + 12right) sin^{2}{left (x right )} cos^{3}{left (x right )}$$
Комбинаторика

3 2 / 4 4 2 2
cos (x)*sin (x)*12*cos (x) + 20*sin (x) – 49*cos (x)*sin (x)/

$$left(20 sin^{4}{left (x right )} – 49 sin^{2}{left (x right )} cos^{2}{left (x right )} + 12 cos^{4}{left (x right )}right) sin^{2}{left (x right )} cos^{3}{left (x right )}$$
Раскрыть выражение

3 2 / 4 4 2 2
cos (x)*sin (x)*12*cos (x) + 20*sin (x) – 49*cos (x)*sin (x)/

$$left(20 sin^{4}{left (x right )} – 49 sin^{2}{left (x right )} cos^{2}{left (x right )} + 12 cos^{4}{left (x right )}right) sin^{2}{left (x right )} cos^{3}{left (x right )}$$
   
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