Question

Помогите с третьим или вторым

Answer 1

2.

$\cos( \frac{\pi}{4} - \beta ) \times \cos( \frac{\pi}{4} + \beta ) = \\ = ( \cos( \frac{\pi}{4} ) \cos( \beta ) + \sin( \frac{\pi}{4} ) \sin( \beta ) ) \times ( \cos( \frac{\pi}{4} ) \cos( \beta ) - \sin( \frac{\pi}{4} ) \sin( \beta ) ) = \\ = ( \frac{ \sqrt{2} }{2} \cos( \beta ) + \frac{ \sqrt{2} }{2} \sin( \beta ) ) \times ( \frac{ \sqrt{2} }{2} \cos( \beta ) - \frac{ \sqrt{2} }{2} \sin( \beta ) ) = \\ = ( \frac{ \sqrt{2} }{2} \cos( \beta )) {}^{2} - ( \frac{ \sqrt{2} }{2} \sin( \beta ) ) {}^{2} = \frac{1}{2} \cos {}^{2} ( \beta ) - \frac{1}{2} \sin {}^{2} ( \beta ) = \frac{1}{2} \cos( 2\beta )$

3.

$\frac{ \cos( \beta ) + \cos( 5\beta ) + \cos( 3\beta ) }{ \sin( \beta ) + \sin( 3\beta ) + \sin( 5\beta ) } = \\\\ = \frac{2 \cos( \frac{ \beta + 5 \beta }{2} ) \cos( \frac{ \beta - 5 \beta }{2} ) + \cos( 3\beta ) }{2 \sin( \frac{ \beta + 5 \beta }{2} ) \cos( \frac{ \beta - 5 \beta }{2} ) + \sin(3 \beta ) } =\\ \\ = \frac{2 \cos( 3\beta ) \cos(2 \beta ) + \cos( 3\beta ) }{2 \sin( 3\beta ) \cos( 2\beta ) + \sin(3 \beta ) } = \\\\ = \frac{ \cos( 3\beta )(2 \cos( 2\beta ) + 1) }{ \sin( 3\beta ) (2 \cos( 2\beta ) + 1)} = tg(3 \beta )$