Дано
$$sin{left (4 x right )} – frac{cos{left (4 x right )}}{- sin{left (2 x right )} + cos{left (2 x right )}}$$
Подстановка условия
$$sin{left (4 x right )} – frac{cos{left (4 x right )}}{- sin{left (2 x right )} + cos{left (2 x right )}}$$
sin(4*(-1/4)) – cos(4*(-1/4))/(cos(2*(-1/4)) – sin(2*(-1/4)))
$$sin{left (4 (-1/4) right )} – frac{cos{left (4 (-1/4) right )}}{- sin{left (2 (-1/4) right )} + cos{left (2 (-1/4) right )}}$$
sin(4*(-1)/4) – cos(4*(-1)/4)/(cos(2*(-1)/4) – sin(2*(-1)/4))
$$sin{left (frac{-4}{4} right )} – frac{cos{left (frac{-4}{4} right )}}{- sin{left (frac{-2}{4} right )} + cos{left (frac{-2}{4} right )}}$$
-sin(1) – cos(1)/(cos(1/2) + sin(1/2))
$$- sin{left (1 right )} – frac{cos{left (1 right )}}{sin{left (frac{1}{2} right )} + cos{left (frac{1}{2} right )}}$$
Численный ответ
-cos(4*x)/(-sin(2*x) + cos(2*x)) + sin(4*x)
Рациональный знаменатель
$$frac{left(- sin{left (2 x right )} + cos{left (2 x right )}right) sin{left (4 x right )} – cos{left (4 x right )}}{- sin{left (2 x right )} + cos{left (2 x right )}}$$
Объединение рациональных выражений
$$frac{left(- sin{left (2 x right )} + cos{left (2 x right )}right) sin{left (4 x right )} – cos{left (4 x right )}}{- sin{left (2 x right )} + cos{left (2 x right )}}$$
Общее упрощение
___
/ 2 *cos(4*x)
– ————— + sin(4*x)
/ pi
2*cos|2*x + –|
4 /
$$sin{left (4 x right )} – frac{sqrt{2} cos{left (4 x right )}}{2 cos{left (2 x + frac{pi}{4} right )}}$$
Соберем выражение
$$sin{left (4 x right )} + frac{cos{left (4 x right )}}{sin{left (2 x right )} – cos{left (2 x right )}}$$
Тригонометрическая часть
___
/ 2 *cos(4*x)
– ————— + sin(4*x)
/ pi
2*cos|2*x + –|
4 /
$$sin{left (4 x right )} – frac{sqrt{2} cos{left (4 x right )}}{2 cos{left (2 x + frac{pi}{4} right )}}$$
Комбинаторика
-cos(4*x) + cos(2*x)*sin(4*x) – sin(2*x)*sin(4*x)
————————————————-
-sin(2*x) + cos(2*x)
$$frac{1}{- sin{left (2 x right )} + cos{left (2 x right )}} left(- sin{left (2 x right )} sin{left (4 x right )} + sin{left (4 x right )} cos{left (2 x right )} – cos{left (4 x right )}right)$$
Раскрыть выражение
4 4 2 2
– cos (x) – sin (x) + 6*cos (x)*sin (x) 3 3
————————————— – 4*sin (x)*cos(x) + 4*cos (x)*sin(x)
2 2
cos (x) – sin (x) – 2*cos(x)*sin(x)
$$- 4 sin^{3}{left (x right )} cos{left (x right )} + 4 sin{left (x right )} cos^{3}{left (x right )} + frac{- sin^{4}{left (x right )} + 6 sin^{2}{left (x right )} cos^{2}{left (x right )} – cos^{4}{left (x right )}}{- sin^{2}{left (x right )} – 2 sin{left (x right )} cos{left (x right )} + cos^{2}{left (x right )}}$$
