Докажите тождество 40БАЛЛОВ(sin(п/2-6а)-cos(п+4а))(sin(п-6а)-cos(3п/2+4а))/1+соs(2п+10a)=sin2a. Created by raptor777novo. algebra-ru

1+соs(2п+10a)=sin2a...

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July 2022 1 6 Report

Докажите тождество 40БАЛЛОВ
(sin(п/2-6а)-cos(п+4а))(sin(п-6а)-cos(3п/2+4а))/1+соs(2п+10a)=sin2a

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NNNLLL54

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Ответ.

Сначала воспользуемся формулами приведения.

$\displaystyle \frac{(\, sin(\frac{\pi}{2}-6a)-cos(\pi +4a))(sin(\pi -6a)-cos(\frac{3\pi}{2}+4a)\, )}{1+cos(2\pi +10a)}=\\\\\\=\frac{(cos6a+cos4a)(sin6a-sin4a)}{1+cos10a}=\frac{2\, cos5a\cdot cosa\cdot \, 2\, sina\cdot cos5a}{2cos^25a}=\\\\\\=\frac{2cos^25a\cdot 2\, sina\cdot cosa}{2cos^25a}=2\, sina\cdot cosa=sin2a\\\\\\sin2a=sin2a$

$\star \ sin2x=2\, sinx\cdot cosx\\\\cosx+cosy=2\cdot cos\dfrac{x+y}{2}\cdot sin\dfrac{x-y}{2}\\\\\\sinx-siny=2\cdot sin\dfrac{x-y}{2}\cdot cos\dfrac{x+y}{2}\ \ \star$

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