Дано
$$frac{2 cos{left (5 x right )}}{cos{left (10 x right )}} sin{left (5 x right )} + frac{2 sin^{2}{left (5 x right )}}{cos^{2}{left (10 x right )}} sin{left (10 x right )}$$
Подстановка условия
$$frac{2 cos{left (5 x right )}}{cos{left (10 x right )}} sin{left (5 x right )} + frac{2 sin^{2}{left (5 x right )}}{cos^{2}{left (10 x right )}} sin{left (10 x right )}$$
((2*cos(5*(1/4)))*sin(5*(1/4)))/cos(10*(1/4)) + ((2*sin(5*(1/4))^2)*sin(10*(1/4)))/cos(10*(1/4))^2
$$frac{2 cos{left (5 (1/4) right )}}{cos{left (10 (1/4) right )}} sin{left (5 (1/4) right )} + frac{2 sin^{2}{left (5 (1/4) right )}}{cos^{2}{left (10 (1/4) right )}} sin{left (10 (1/4) right )}$$
((2*cos(5/4))*sin(5/4))/cos(10/4) + ((2*sin(5/4)^2)*sin(10/4))/cos(10/4)^2
$$frac{2 cos{left (frac{5}{4} right )}}{cos{left (frac{10}{4} right )}} sin{left (frac{5}{4} right )} + frac{2 sin^{2}{left (frac{5}{4} right )}}{cos^{2}{left (frac{10}{4} right )}} sin{left (frac{10}{4} right )}$$
2*cos(5/4)*sin(5/4)/cos(5/2) + 2*sin(5/4)^2*sin(5/2)/cos(5/2)^2
$$frac{2 cos{left (frac{5}{4} right )}}{cos{left (frac{5}{2} right )}} sin{left (frac{5}{4} right )} + frac{2 sin^{2}{left (frac{5}{4} right )}}{cos^{2}{left (frac{5}{2} right )}} sin{left (frac{5}{2} right )}$$
Степени
$$frac{2 sin^{2}{left (5 x right )}}{cos^{2}{left (10 x right )}} sin{left (10 x right )} + frac{2 cos{left (5 x right )}}{cos{left (10 x right )}} sin{left (5 x right )}$$
Численный ответ
2.0*cos(5*x)*sin(5*x)/cos(10*x) + 2.0*sin(5*x)^2*sin(10*x)/cos(10*x)^2
Рациональный знаменатель
$$frac{1}{cos^{3}{left (10 x right )}} left(2 sin^{2}{left (5 x right )} sin{left (10 x right )} cos{left (10 x right )} + 2 sin{left (5 x right )} cos{left (5 x right )} cos^{2}{left (10 x right )}right)$$
Объединение рациональных выражений
$$frac{2 sin{left (5 x right )}}{cos^{2}{left (10 x right )}} left(sin{left (5 x right )} sin{left (10 x right )} + cos{left (5 x right )} cos{left (10 x right )}right)$$
Общее упрощение
2*sin(10*x)
————-
1 + cos(20*x)
$$frac{2 sin{left (10 x right )}}{cos{left (20 x right )} + 1}$$
Соберем выражение
$$left(- cos{left (10 x right )} + 1right) tan{left (10 x right )} sec{left (10 x right )} + tan{left (10 x right )}$$
Общий знаменатель
2
2*sin (5*x)*sin(10*x) + 2*cos(5*x)*cos(10*x)*sin(5*x)
—————————————————–
2
cos (10*x)
$$frac{1}{cos^{2}{left (10 x right )}} left(2 sin^{2}{left (5 x right )} sin{left (10 x right )} + 2 sin{left (5 x right )} cos{left (5 x right )} cos{left (10 x right )}right)$$
Тригонометрическая часть
2*sin(10*x)
————-
1 + cos(20*x)
$$frac{2 sin{left (10 x right )}}{cos{left (20 x right )} + 1}$$
Комбинаторика
2*(cos(5*x)*cos(10*x) + sin(5*x)*sin(10*x))*sin(5*x)
—————————————————-
2
cos (10*x)
$$frac{2 sin{left (5 x right )}}{cos^{2}{left (10 x right )}} left(sin{left (5 x right )} sin{left (10 x right )} + cos{left (5 x right )} cos{left (10 x right )}right)$$
Раскрыть выражение
2
/ 5 2 3 4 / 5 3 2 4 / 5 2 3 4 / 3 7 7 3 9 9 5 5
sin (x) – 10*cos (x)*sin (x) + 5*cos (x)*sin(x)/*2*cos (x) – 20*cos (x)*sin (x) + 10*sin (x)*cos(x)/ 2*sin (x) – 10*cos (x)*sin (x) + 5*cos (x)*sin(x)/ * – 120*cos (x)*sin (x) – 120*cos (x)*sin (x) + 10*cos (x)*sin(x) + 10*sin (x)*cos(x) + 252*cos (x)*sin (x)/
——————————————————————————————————— + —————————————————————————————————————————————————————-
10 10 4 6 8 2 2 8 6 4 2
cos (x) – sin (x) – 210*cos (x)*sin (x) – 45*cos (x)*sin (x) + 45*cos (x)*sin (x) + 210*cos (x)*sin (x) / 10 10 4 6 8 2 2 8 6 4
cos (x) – sin (x) – 210*cos (x)*sin (x) – 45*cos (x)*sin (x) + 45*cos (x)*sin (x) + 210*cos (x)*sin (x)/
$$frac{left(10 sin^{4}{left (x right )} cos{left (x right )} – 20 sin^{2}{left (x right )} cos^{3}{left (x right )} + 2 cos^{5}{left (x right )}right) left(sin^{5}{left (x right )} – 10 sin^{3}{left (x right )} cos^{2}{left (x right )} + 5 sin{left (x right )} cos^{4}{left (x right )}right)}{- sin^{10}{left (x right )} + 45 sin^{8}{left (x right )} cos^{2}{left (x right )} – 210 sin^{6}{left (x right )} cos^{4}{left (x right )} + 210 sin^{4}{left (x right )} cos^{6}{left (x right )} – 45 sin^{2}{left (x right )} cos^{8}{left (x right )} + cos^{10}{left (x right )}} + frac{2 left(sin^{5}{left (x right )} – 10 sin^{3}{left (x right )} cos^{2}{left (x right )} + 5 sin{left (x right )} cos^{4}{left (x right )}right)^{2} left(10 sin^{9}{left (x right )} cos{left (x right )} – 120 sin^{7}{left (x right )} cos^{3}{left (x right )} + 252 sin^{5}{left (x right )} cos^{5}{left (x right )} – 120 sin^{3}{left (x right )} cos^{7}{left (x right )} + 10 sin{left (x right )} cos^{9}{left (x right )}right)}{left(- sin^{10}{left (x right )} + 45 sin^{8}{left (x right )} cos^{2}{left (x right )} – 210 sin^{6}{left (x right )} cos^{4}{left (x right )} + 210 sin^{4}{left (x right )} cos^{6}{left (x right )} – 45 sin^{2}{left (x right )} cos^{8}{left (x right )} + cos^{10}{left (x right )}right)^{2}}$$
