Question

Sa se arate ca
$\cos( \alpha ) + \sin( \alpha ) = \sqrt{2} \cos( \alpha - \frac{\pi}{4} )$
oricare ar fi
$\alpha$
apartine lui R. ​

Answer 1

$\cos\alpha +\sin \alpha = \sin\Big(\dfrac{\pi}{2}-\alpha\Big)+\sin\alpha =\\\\ \\ =2\sin \left(\dfrac{\frac{\pi}{2}-\alpha +\alpha }{2}\right)\cos\left(\dfrac{\frac{\pi}{2}-\alpha - \alpha}{2}\right) = \\ \\ =2\sin \left(\dfrac{\pi}{4}\right)\cos \left(\dfrac{\frac{\pi}{2}-2\alpha}{2}\right) =\\ \\ =2\cdot \dfrac{\sqrt 2}{2}\cdot \cos \left(\dfrac{\pi}{4}-\alpha\right) =\\ \\ \\=\boxed{\sqrt{2}\cos\left(\alpha -\dfrac{\pi}{4}\right)}$