cos(pi*1/7+a)*cos(5*pi*1/14-a)-sin(pi*1/7+a)*sin(5*pi*1/14) если a=1/4 (упростите выражение)

Дано

$$- \sin{\left (\frac{5 \pi}{14} \right )} \sin{\left (a + \frac{\pi}{7} \right )} + \cos{\left (- a + \frac{5 \pi}{14} \right )} \cos{\left (a + \frac{\pi}{7} \right )}$$
Подстановка условия
$$- \sin{\left (\frac{5 \pi}{14} \right )} \sin{\left (a + \frac{\pi}{7} \right )} + \cos{\left (- a + \frac{5 \pi}{14} \right )} \cos{\left (a + \frac{\pi}{7} \right )}$$

cos(pi/7 + (1/4))*cos((5*pi)/14 — (1/4)) — sin(pi/7 + (1/4))*sin((5*pi)/14)

$$- \sin{\left (\frac{5 \pi}{14} \right )} \sin{\left ((1/4) + \frac{\pi}{7} \right )} + \cos{\left (- (1/4) + \frac{5 \pi}{14} \right )} \cos{\left ((1/4) + \frac{\pi}{7} \right )}$$

cos(pi/7 + 1/4)*cos((5*pi)/14 — 1/4) — sin(pi/7 + 1/4)*sin((5*pi)/14)

$$- \sin{\left (\frac{5 \pi}{14} \right )} \sin{\left (\frac{1}{4} + \frac{\pi}{7} \right )} + \cos{\left (\frac{1}{4} + \frac{\pi}{7} \right )} \cos{\left (- \frac{1}{4} + \frac{5 \pi}{14} \right )}$$

cos(1/4 + pi/7)*sin(1/4 + pi/7) — sin(5*pi/14)*sin(1/4 + pi/7)

$$- \sin{\left (\frac{5 \pi}{14} \right )} \sin{\left (\frac{1}{4} + \frac{\pi}{7} \right )} + \sin{\left (\frac{1}{4} + \frac{\pi}{7} \right )} \cos{\left (\frac{1}{4} + \frac{\pi}{7} \right )}$$
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Степени
$$\sin{\left (a + \frac{\pi}{7} \right )} \cos{\left (a + \frac{\pi}{7} \right )} — \sin{\left (\frac{5 \pi}{14} \right )} \sin{\left (a + \frac{\pi}{7} \right )}$$
Численный ответ

-0.900968867902419*sin(pi/7 + a) + cos(pi/7 + a)*cos((5*pi)/14 — a)

Рациональный знаменатель
$$\sin{\left (a + \frac{\pi}{7} \right )} \cos{\left (a + \frac{\pi}{7} \right )} — \sin{\left (\frac{5 \pi}{14} \right )} \sin{\left (a + \frac{\pi}{7} \right )}$$
Объединение рациональных выражений
$$- \sin{\left (\frac{5 \pi}{14} \right )} \sin{\left (\frac{1}{7} \left(7 a + \pi\right) \right )} + \cos{\left (\frac{1}{14} \left(- 14 a + 5 \pi\right) \right )} \cos{\left (\frac{1}{7} \left(7 a + \pi\right) \right )}$$
Общее упрощение

/ /pi / pi\ / pi
|- cos|—| + cos|a + —||*sin|a + —|
7 / 7 // 7 /

$$\left(\cos{\left (a + \frac{\pi}{7} \right )} — \cos{\left (\frac{\pi}{7} \right )}\right) \sin{\left (a + \frac{\pi}{7} \right )}$$
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Соберем выражение
$$- \frac{1}{2} \sin{\left (a \right )} — \frac{1}{2} \sin{\left (a + \frac{2 \pi}{7} \right )} + \frac{1}{2} \sin{\left (2 a + \frac{2 \pi}{7} \right )}$$
Комбинаторика

/ /5*pi / pi\ / pi
|- sin|—-| + cos|a + —||*sin|a + —|
14 / 7 // 7 /

$$\left(\cos{\left (a + \frac{\pi}{7} \right )} — \sin{\left (\frac{5 \pi}{14} \right )}\right) \sin{\left (a + \frac{\pi}{7} \right )}$$
Общий знаменатель

/ pi / pi /5*pi / pi
cos|a + —|*sin|a + —| — sin|—-|*sin|a + —|
7 / 7 / 14 / 7 /

$$\sin{\left (a + \frac{\pi}{7} \right )} \cos{\left (a + \frac{\pi}{7} \right )} — \sin{\left (\frac{5 \pi}{14} \right )} \sin{\left (a + \frac{\pi}{7} \right )}$$
Тригонометрическая часть

/ / pi\ / 2*pi
sin|2*|a + —|| sin|a + —-|
7 // sin(a) 7 /
————— — —— — ————-
2 2 2

$$- \frac{1}{2} \sin{\left (a \right )} + \frac{1}{2} \sin{\left (2 \left(a + \frac{\pi}{7}\right) \right )} — \frac{1}{2} \sin{\left (a + \frac{2 \pi}{7} \right )}$$
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Раскрыть выражение

/ /pi /pi\ / /5*pi /5*pi\ / /pi /pi /5*pi
|cos(a)*cos|—| — sin(a)*sin|—||*|cos(a)*cos|—-| + sin(a)*sin|—-|| — |cos(a)*sin|—| + cos|—|*sin(a)|*sin|—-|
7 / 7 // 14 / 14 // 7 / 7 / / 14 /

$$\left(- \sin{\left (\frac{\pi}{7} \right )} \sin{\left (a \right )} + \cos{\left (\frac{\pi}{7} \right )} \cos{\left (a \right )}\right) \left(\sin{\left (\frac{5 \pi}{14} \right )} \sin{\left (a \right )} + \cos{\left (\frac{5 \pi}{14} \right )} \cos{\left (a \right )}\right) — \left(\sin{\left (a \right )} \cos{\left (\frac{\pi}{7} \right )} + \sin{\left (\frac{\pi}{7} \right )} \cos{\left (a \right )}\right) \sin{\left (\frac{5 \pi}{14} \right )}$$
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