Дано
$$frac{1}{5} left(-1 cdot 2 sin{left (t right )}right) left(p – frac{4}{5} sin{left (t right )}right) + frac{8 cos^{2}{left (t right )}}{25 left(p – frac{4}{5} sin{left (t right )}right)^{2}}$$
Подстановка условия
$$frac{1}{5} left(-1 cdot 2 sin{left (t right )}right) left(p – frac{4}{5} sin{left (t right )}right) + frac{8 cos^{2}{left (t right )}}{25 left(p – frac{4}{5} sin{left (t right )}right)^{2}}$$
((-2*sin(t))/5)*((-2) – 4*sin(t)/5) + (8*cos(t)^2)/(25*((-2) – 4*sin(t)/5)^2)
$$frac{1}{5} left(-1 cdot 2 sin{left (t right )}right) left((-2) – frac{4}{5} sin{left (t right )}right) + frac{8 cos^{2}{left (t right )}}{25 left((-2) – frac{4}{5} sin{left (t right )}right)^{2}}$$
((-2*sin(t))/5)*(-2 – 4*sin(t)/5) + (8*cos(t)^2)/(25*(-2 – 4*sin(t)/5)^2)
$$frac{1}{5} left(-1 cdot 2 sin{left (t right )}right) left(- frac{4}{5} sin{left (t right )} – 2right) + frac{8 cos^{2}{left (t right )}}{25 left(- frac{4}{5} sin{left (t right )} – 2right)^{2}}$$
-2*(-2 – 4*sin(t)/5)*sin(t)/5 + 8*cos(t)^2/(25*(-2 – 4*sin(t)/5)^2)
$$- frac{2}{5} left(- frac{4}{5} sin{left (t right )} – 2right) sin{left (t right )} + frac{8 cos^{2}{left (t right )}}{25 left(- frac{4}{5} sin{left (t right )} – 2right)^{2}}$$
Степени
$$- frac{2}{5} left(p – frac{4}{5} sin{left (t right )}right) sin{left (t right )} + frac{8 cos^{2}{left (t right )}}{25 left(p – frac{4}{5} sin{left (t right )}right)^{2}}$$
Численный ответ
0.32*cos(t)^2/(p – 0.8*sin(t))^2 – 0.4*(p – 0.8*sin(t))*sin(t)
Рациональный знаменатель
$$frac{1}{625 left(5 p – 4 sin{left (t right )}right)^{2}} left(- 50 left(5 p – 4 sin{left (t right )}right)^{3} sin{left (t right )} + 5000 cos^{2}{left (t right )}right)$$
Объединение рациональных выражений
$$frac{1}{25 left(5 p – 4 sin{left (t right )}right)^{2}} left(- 2 left(5 p – 4 sin{left (t right )}right)^{3} sin{left (t right )} + 200 cos^{2}{left (t right )}right)$$
Общее упрощение
/ 2 3
2*100*cos (t) – (-4*sin(t) + 5*p) *sin(t)/
——————————————-
2
25*(-4*sin(t) + 5*p)
$$frac{1}{25 left(5 p – 4 sin{left (t right )}right)^{2}} left(- 2 left(5 p – 4 sin{left (t right )}right)^{3} sin{left (t right )} + 200 cos^{2}{left (t right )}right)$$
Соберем выражение
$$- frac{2 p}{5} sin{left (t right )} + frac{- 4 cos{left (2 t right )} – 4}{- 25 p^{2} + 40 p sin{left (t right )} + 8 cos{left (2 t right )} – 8} – frac{4}{25} cos{left (2 t right )} + frac{4}{25}$$
Общий знаменатель
2 2
8*sin (t) 8*cos (t) 2*p*sin(t)
——— + ——————————– – ———-
25 2 2 5
16*sin (t) + 25*p – 40*p*sin(t)
$$- frac{2 p}{5} sin{left (t right )} + frac{8}{25} sin^{2}{left (t right )} + frac{8 cos^{2}{left (t right )}}{25 p^{2} – 40 p sin{left (t right )} + 16 sin^{2}{left (t right )}}$$
Тригонометрическая часть
/ 4*sin(t)
2*|p – ——–|*sin(t) 2
5 / 8*cos (t)
– ———————– + ——————
5 2
/ 4*sin(t)
25*|p – ——–|
5 /
$$- frac{2}{5} left(p – frac{4}{5} sin{left (t right )}right) sin{left (t right )} + frac{8 cos^{2}{left (t right )}}{25 left(p – frac{4}{5} sin{left (t right )}right)^{2}}$$
Комбинаторика
/ 2 4 2 2 3 3
-2* – 100*cos (t) – 64*sin (t) – 300*p *sin (t) + 125*p *sin(t) + 240*p*sin (t)/
——————————————————————————–
2
25*(-4*sin(t) + 5*p)
$$- frac{1}{25 left(5 p – 4 sin{left (t right )}right)^{2}} left(250 p^{3} sin{left (t right )} – 600 p^{2} sin^{2}{left (t right )} + 480 p sin^{3}{left (t right )} – 128 sin^{4}{left (t right )} – 200 cos^{2}{left (t right )}right)$$
Раскрыть выражение
/ 8*sin(t)
|2*p – ——–|*sin(t) 2
5 / 8*cos (t)
– ———————– + ——————
5 2
/ 4*sin(t)
25*|p – ——–|
5 /
$$- frac{1}{5} left(2 p – frac{8}{5} sin{left (t right )}right) sin{left (t right )} + frac{8 cos^{2}{left (t right )}}{25 left(p – frac{4}{5} sin{left (t right )}right)^{2}}$$
