Question

Prove $\sqrt{2}.\cos (\frac{\pi }{4}+A)=\cos A-\sin A$

Answer 1

We know that

$\cos (A+B)=\cos A\cos B-\sin A\sin B$

$\cos (\frac{\pi }{4}+A)=\cos \frac{\pi }{4}\cos A-\sin \frac{\pi }{4}\sin A$

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

$\sqrt{2}\cos (\frac{\pi }{4}+A)=\cos A-\sin A$

Hence proved