Дано
$$- sin^{2}{left (a + frac{15 pi}{8} right )} + cos^{2}{left (a + frac{5 pi}{8} right )}$$
Подстановка условия
$$- sin^{2}{left (a + frac{15 pi}{8} right )} + cos^{2}{left (a + frac{5 pi}{8} right )}$$
cos((5*pi)/8 + (1/4))^2 – sin((15*pi)/8 + (1/4))^2
$$- sin^{2}{left ((1/4) + frac{15 pi}{8} right )} + cos^{2}{left ((1/4) + frac{5 pi}{8} right )}$$
cos((5*pi)/8 + 1/4)^2 – sin((15*pi)/8 + 1/4)^2
$$- sin^{2}{left (frac{1}{4} + frac{15 pi}{8} right )} + cos^{2}{left (frac{1}{4} + frac{5 pi}{8} right )}$$
sin(1/4 + pi/8)^2 – cos(1/4 + 3*pi/8)^2
$$- cos^{2}{left (frac{1}{4} + frac{3 pi}{8} right )} + sin^{2}{left (frac{1}{4} + frac{pi}{8} right )}$$
Степени
$$sin^{2}{left (a + frac{pi}{8} right )} – cos^{2}{left (a + frac{3 pi}{8} right )}$$
Численный ответ
cos((5*pi)/8 + a)^2 – sin((15*pi)/8 + a)^2
Рациональный знаменатель
$$sin^{2}{left (a + frac{pi}{8} right )} – cos^{2}{left (a + frac{3 pi}{8} right )}$$
Объединение рациональных выражений
$$- sin^{2}{left (frac{1}{8} left(8 a + 15 piright) right )} + cos^{2}{left (frac{1}{8} left(8 a + 5 piright) right )}$$
Общее упрощение
___
/ 2 *sin(2*a)
————–
2
$$frac{sqrt{2}}{2} sin{left (2 a right )}$$
Соберем выражение
$$frac{1}{2} sin{left (2 a + frac{pi}{4} right )} – frac{1}{2} cos{left (2 a + frac{pi}{4} right )}$$
Общий знаменатель
2/ pi 2/ 3*pi
sin |a + –| – cos |a + —-|
8 / 8 /
$$sin^{2}{left (a + frac{pi}{8} right )} – cos^{2}{left (a + frac{3 pi}{8} right )}$$
Тригонометрическая часть
___________ ___________
/ ___ / ___
/ 2 + / 2 */ 2 – / 2 *sin(2*a)
————————————–
2
$$frac{1}{2} sqrt{- sqrt{2} + 2} sqrt{sqrt{2} + 2} sin{left (2 a right )}$$
Комбинаторика
/ / pi / 3*pi / / 3*pi / pi
-|- sin|a + –| + cos|a + —-||*|cos|a + —-| + sin|a + –||
8 / 8 // 8 / 8 //
$$- left(- sin{left (a + frac{pi}{8} right )} + cos{left (a + frac{3 pi}{8} right )}right) left(sin{left (a + frac{pi}{8} right )} + cos{left (a + frac{3 pi}{8} right )}right)$$
Раскрыть выражение
2 2
/ ___________ ___________ / ___________ ___________
| / ___ / ___ | | / ___ / ___ |
| / 1 / 2 / 1 / 2 | | / 1 / 2 / 1 / 2 |
|- / – – —– *cos(a) – / – + —– *sin(a)| – | / – + —– *sin(a) – / – – —– *cos(a)|
/ 2 4 / 2 4 / / 2 4 / 2 4 /
$$left(- sqrt{frac{sqrt{2}}{4} + frac{1}{2}} sin{left (a right )} – sqrt{- frac{sqrt{2}}{4} + frac{1}{2}} cos{left (a right )}right)^{2} – left(sqrt{frac{sqrt{2}}{4} + frac{1}{2}} sin{left (a right )} – sqrt{- frac{sqrt{2}}{4} + frac{1}{2}} cos{left (a right )}right)^{2}$$
