Question

Prove the following :
$\sqrt{2}\cdot \cos (\frac{\pi }{4}+A)=\cos A-\sin A$

Answer 1

we know $\sin (a+b)=\sin \left(a\right)\cos \left(b\right)+\cos \left(a\right)\sin \left(b\right)$

given $\sqrt{2}\sin (\frac{\pi }{4}+A)$

$\Rightarrow \sqrt{2}(\sin \frac{\pi }{4}\cos A+\cos \frac{\pi }{4}\sin A)=\sqrt{2}(\frac{\cos A}{\sqrt{2}}+\frac{\sin A}{\sqrt{2}})=\cos A+\sin A$[since by above formula]