Дано
$$frac{1}{left(5 cos{left (x right )} – 4right)^{2}} left(-1 cdot 45 left(cos{left (x right )} + frac{10 sin^{2}{left (x right )}}{5 cos{left (x right )} – 4}right)right)$$
Подстановка условия
$$frac{1}{left(5 cos{left (x right )} – 4right)^{2}} left(-1 cdot 45 left(cos{left (x right )} + frac{10 sin^{2}{left (x right )}}{5 cos{left (x right )} – 4}right)right)$$
(-45*((10*sin((1/2))^2)/(-4 + 5*cos((1/2))) + cos((1/2))))/(-4 + 5*cos((1/2)))^2
$$frac{1}{left(5 cos{left ((1/2) right )} – 4right)^{2}} left(-1 cdot 45 left(cos{left ((1/2) right )} + frac{10 sin^{2}{left ((1/2) right )}}{5 cos{left ((1/2) right )} – 4}right)right)$$
(-45*((10*sin(1/2)^2)/(-4 + 5*cos(1/2)) + cos(1/2)))/(-4 + 5*cos(1/2))^2
$$frac{1}{left(-4 + 5 cos{left (frac{1}{2} right )}right)^{2}} left(-1 cdot 45 left(cos{left (frac{1}{2} right )} + frac{10 sin^{2}{left (frac{1}{2} right )}}{-4 + 5 cos{left (frac{1}{2} right )}}right)right)$$
(-45*cos(1/2) – 450*sin(1/2)^2/(-4 + 5*cos(1/2)))/(-4 + 5*cos(1/2))^2
$$frac{1}{left(-4 + 5 cos{left (frac{1}{2} right )}right)^{2}} left(- frac{450 sin^{2}{left (frac{1}{2} right )}}{-4 + 5 cos{left (frac{1}{2} right )}} – 45 cos{left (frac{1}{2} right )}right)$$
Степени
$$frac{1}{left(5 cos{left (x right )} – 4right)^{2}} left(- 45 cos{left (x right )} – frac{450 sin^{2}{left (x right )}}{5 cos{left (x right )} – 4}right)$$
Численный ответ
-45.0*(10.0*sin(x)^2/(-4.0 + 5.0*cos(x)) + cos(x))/(-4.0 + 5.0*cos(x))^2
Рациональный знаменатель
$$frac{1}{left(5 cos{left (x right )} – 4right)^{3}} left(- 45 left(5 cos{left (x right )} – 4right) cos{left (x right )} – 450 sin^{2}{left (x right )}right)$$
Объединение рациональных выражений
$$- frac{1}{left(5 cos{left (x right )} – 4right)^{3}} left(45 left(5 cos{left (x right )} – 4right) cos{left (x right )} + 450 sin^{2}{left (x right )}right)$$
Общее упрощение
/ 2
45* -5 – 5*sin (x) + 4*cos(x)/
——————————
3
(-4 + 5*cos(x))
$$frac{- 225 sin^{2}{left (x right )} + 180 cos{left (x right )} – 225}{left(5 cos{left (x right )} – 4right)^{3}}$$
Соберем выражение
$$frac{1}{left(5 cos{left (x right )} – 4right)^{2}} left(- 45 cos{left (x right )} – frac{450 sin^{2}{left (x right )}}{5 cos{left (x right )} – 4}right)$$
1350 450*cos(2*x) 720*cos(x)
– ———————————————— + ———————————————— + ————————————————
-856 – 600*cos(2*x) + 125*cos(3*x) + 1335*cos(x) -856 – 600*cos(2*x) + 125*cos(3*x) + 1335*cos(x) -856 – 600*cos(2*x) + 125*cos(3*x) + 1335*cos(x)
$$frac{720 cos{left (x right )}}{1335 cos{left (x right )} – 600 cos{left (2 x right )} + 125 cos{left (3 x right )} – 856} + frac{450 cos{left (2 x right )}}{1335 cos{left (x right )} – 600 cos{left (2 x right )} + 125 cos{left (3 x right )} – 856} – frac{1350}{1335 cos{left (x right )} – 600 cos{left (2 x right )} + 125 cos{left (3 x right )} – 856}$$
Общий знаменатель
/ 2 2
– -180*cos(x) + 225*cos (x) + 450*sin (x)/
——————————————–
2 3
-64 – 300*cos (x) + 125*cos (x) + 240*cos(x)
$$- frac{450 sin^{2}{left (x right )} + 225 cos^{2}{left (x right )} – 180 cos{left (x right )}}{125 cos^{3}{left (x right )} – 300 cos^{2}{left (x right )} + 240 cos{left (x right )} – 64}$$
Комбинаторика
/ 2 2
-45* -4*cos(x) + 5*cos (x) + 10*sin (x)/
—————————————-
3
(-4 + 5*cos(x))
$$- frac{1}{left(5 cos{left (x right )} – 4right)^{3}} left(450 sin^{2}{left (x right )} + 225 cos^{2}{left (x right )} – 180 cos{left (x right )}right)$$
Тригонометрическая часть
/ 2
45* -5 – 5*sin (x) + 4*cos(x)/
——————————
3
(-4 + 5*cos(x))
$$frac{- 225 sin^{2}{left (x right )} + 180 cos{left (x right )} – 225}{left(5 cos{left (x right )} – 4right)^{3}}$$
Раскрыть выражение
2
450*sin (x)
-45*cos(x) – ————-
-4 + 5*cos(x)
————————–
2
(-4 + 5*cos(x))
$$frac{1}{left(5 cos{left (x right )} – 4right)^{2}} left(- 45 cos{left (x right )} – frac{450 sin^{2}{left (x right )}}{5 cos{left (x right )} – 4}right)$$
