Дано
$$cos^{2}{left (- frac{x}{4} + frac{3 pi}{8} right )} – cos^{2}{left (frac{x}{4} + frac{11 pi}{8} right )}$$
Подстановка условия
$$cos^{2}{left (- frac{x}{4} + frac{3 pi}{8} right )} – cos^{2}{left (frac{x}{4} + frac{11 pi}{8} right )}$$
cos((3*pi)/8 – (-1/3)/4)^2 – cos((11*pi)/8 + (-1/3)/4)^2
$$cos^{2}{left (- frac{(-1/3)}{4} + frac{3 pi}{8} right )} – cos^{2}{left (frac{(-1/3)}{4} + frac{11 pi}{8} right )}$$
cos((3*pi)/8 – (-1)/(3*4))^2 – cos((11*pi)/8 – 1/12)^2
$$- cos^{2}{left (- frac{1}{12} + frac{11 pi}{8} right )} + cos^{2}{left (- frac{-1}{12} + frac{3 pi}{8} right )}$$
cos(1/12 + 3*pi/8)^2 – sin(1/12 + pi/8)^2
$$- sin^{2}{left (frac{1}{12} + frac{pi}{8} right )} + cos^{2}{left (frac{1}{12} + frac{3 pi}{8} right )}$$
Степени
$$sin^{2}{left (frac{x}{4} + frac{pi}{8} right )} – cos^{2}{left (frac{x}{4} + frac{3 pi}{8} right )}$$
Численный ответ
cos((3*pi)/8 – x/4)^2 – cos((11*pi)/8 + x/4)^2
Рациональный знаменатель
$$sin^{2}{left (frac{x}{4} + frac{pi}{8} right )} – cos^{2}{left (frac{x}{4} + frac{3 pi}{8} right )}$$
Объединение рациональных выражений
$$cos^{2}{left (frac{1}{8} left(- 2 x + 3 piright) right )} – cos^{2}{left (frac{1}{8} left(2 x + 11 piright) right )}$$
Общее упрощение
___ /x
/ 2 *sin|-|
2/
————
2
$$frac{sqrt{2}}{2} sin{left (frac{x}{2} right )}$$
Соберем выражение
$$frac{1}{2} sin{left (frac{x}{2} + frac{pi}{4} right )} – frac{1}{2} cos{left (frac{x}{2} + frac{pi}{4} right )}$$
Общий знаменатель
2/x pi 2/x 3*pi
sin |- + –| – cos |- + —-|
4 8 / 4 8 /
$$sin^{2}{left (frac{x}{4} + frac{pi}{8} right )} – cos^{2}{left (frac{x}{4} + frac{3 pi}{8} right )}$$
Тригонометрическая часть
___________ ___________
/ ___ / ___ /x
/ 2 + / 2 */ 2 – / 2 *sin|-|
2/
————————————
2
$$frac{1}{2} sqrt{- sqrt{2} + 2} sqrt{sqrt{2} + 2} sin{left (frac{x}{2} right )}$$
Комбинаторика
/ /x pi /x 3*pi / /x 3*pi /x pi
-|- sin|- + –| + cos|- + —-||*|cos|- + —-| + sin|- + –||
4 8 / 4 8 // 4 8 / 4 8 //
$$- left(- sin{left (frac{x}{4} + frac{pi}{8} right )} + cos{left (frac{x}{4} + frac{3 pi}{8} right )}right) left(sin{left (frac{x}{4} + frac{pi}{8} right )} + cos{left (frac{x}{4} + frac{3 pi}{8} right )}right)$$
Раскрыть выражение
2 2
/ ___________ ___________ / ___________ ___________
| / ___ / ___ | | / ___ / ___ |
| / 1 / 2 /x / 1 / 2 /x| | / 1 / 2 /x / 1 / 2 /x|
| / – – —– *cos|-| + / – + —– *sin|-|| – | / – + —– *sin|-| – / – – —– *cos|-||
/ 2 4 4/ / 2 4 4// / 2 4 4/ / 2 4 4//
$$- left(sqrt{frac{sqrt{2}}{4} + frac{1}{2}} sin{left (frac{x}{4} right )} – sqrt{- frac{sqrt{2}}{4} + frac{1}{2}} cos{left (frac{x}{4} right )}right)^{2} + left(sqrt{frac{sqrt{2}}{4} + frac{1}{2}} sin{left (frac{x}{4} right )} + sqrt{- frac{sqrt{2}}{4} + frac{1}{2}} cos{left (frac{x}{4} right )}right)^{2}$$
