Question

Доказать тождество:
$\frac{ \sin( \alpha ) + \cos( \alpha ) }{ \sqrt{2} } = \cos( \alpha - \binom{\pi}{4} )$

Answer 1

$\tt\displaystyle \frac{sin(\alpha) + cos(\alpha)}{\sqrt{2}}=cos(\alpha)\cdot cos\bigg(\frac{\pi}{4}\bigg) + sin(\alpha)\cdot sin\bigg(\frac{\pi}{4}\bigg)\\\\\\\frac{sin(\alpha) + cos(\alpha)}{\sqrt{2}} = \frac{\sqrt{2}\cdot cos(\alpha)}{2}+\frac{\sqrt{2}\cdot sin(\alpha)}{2}~~~~~~~~~~~~~~\bigg | \cdot (\sqrt{2})\\\\\\sin(\alpha) + cos(\alpha) = \frac{2\cdot cos(\alpha)}{2} + \frac{2\cdot sin(\alpha)}{2}\\\\\\sin(\alpha) + cos(\alpha) = cos(\alpha) + sin(\alpha)$