Дано
$$frac{sin{left (a + 15 right )} – cos{left (a + 45 right )}}{sin{left (a + 45 right )} + cos{left (a + 45 right )}}$$
Подстановка условия
$$frac{sin{left (a + 15 right )} – cos{left (a + 45 right )}}{sin{left (a + 45 right )} + cos{left (a + 45 right )}}$$
(sin(15 + (1/2)) – cos(45 + (1/2)))/(sin(45 + (1/2)) + cos(45 + (1/2)))
$$frac{sin{left ((1/2) + 15 right )} – cos{left ((1/2) + 45 right )}}{sin{left ((1/2) + 45 right )} + cos{left ((1/2) + 45 right )}}$$
(sin(15 + 1/2) – cos(45 + 1/2))/(sin(45 + 1/2) + cos(45 + 1/2))
$$frac{- cos{left (frac{1}{2} + 45 right )} + sin{left (frac{1}{2} + 15 right )}}{cos{left (frac{1}{2} + 45 right )} + sin{left (frac{1}{2} + 45 right )}}$$
(-cos(91/2) + sin(31/2))/(cos(91/2) + sin(91/2))
$$frac{- cos{left (frac{91}{2} right )} + sin{left (frac{31}{2} right )}}{cos{left (frac{91}{2} right )} + sin{left (frac{91}{2} right )}}$$
Численный ответ
(-cos(45 + a) + sin(15 + a))/(cos(45 + a) + sin(45 + a))
Общее упрощение
___
/ 2 *(-cos(45 + a) + sin(15 + a))
———————————-
/ pi
2*sin|45 + a + –|
4 /
$$frac{sqrt{2} left(sin{left (a + 15 right )} – cos{left (a + 45 right )}right)}{2 sin{left (a + frac{pi}{4} + 45 right )}}$$
Соберем выражение
$$frac{sin{left (a + 15 right )}}{sin{left (a + 45 right )} + cos{left (a + 45 right )}} – frac{cos{left (a + 45 right )}}{sin{left (a + 45 right )} + cos{left (a + 45 right )}}$$
Общий знаменатель
sin(15 + a) + sin(45 + a)
-1 + ————————-
cos(45 + a) + sin(45 + a)
$$frac{sin{left (a + 15 right )} + sin{left (a + 45 right )}}{sin{left (a + 45 right )} + cos{left (a + 45 right )}} – 1$$
Комбинаторика
-(-sin(15 + a) + cos(45 + a))
——————————
cos(45 + a) + sin(45 + a)
$$- frac{- sin{left (a + 15 right )} + cos{left (a + 45 right )}}{sin{left (a + 45 right )} + cos{left (a + 45 right )}}$$
Тригонометрическая часть
___
/ 2 *(sin(15 + a) – cos(45 + a))
———————————
/ pi
2*sin|45 + a + –|
4 /
$$frac{sqrt{2} left(sin{left (a + 15 right )} – cos{left (a + 45 right )}right)}{2 sin{left (a + frac{pi}{4} + 45 right )}}$$
Раскрыть выражение
cos(15)*sin(a) + cos(a)*sin(15) + sin(45)*sin(a) – cos(45)*cos(a)
—————————————————————–
cos(45)*cos(a) + cos(45)*sin(a) + cos(a)*sin(45) – sin(45)*sin(a)
$$frac{sin{left (a right )} cos{left (15 right )} + sin{left (45 right )} sin{left (a right )} – cos{left (45 right )} cos{left (a right )} + sin{left (15 right )} cos{left (a right )}}{- sin{left (45 right )} sin{left (a right )} + sin{left (a right )} cos{left (45 right )} + cos{left (45 right )} cos{left (a right )} + sin{left (45 right )} cos{left (a right )}}$$
