Question

If sinA+cosA=root2cosA then prove cosA-sinA=root2sinA​

Answer 1

Answer:

$\cos A - \sin A = \sqrt{2} \sin A$ (Proved)

Step-by-step explanation:

We are given that $\sin A + \cos A =\sqrt{2} \cos A$ and we have to prove that

$\cos A - \sin A = \sqrt{2} \sin A$.

Now,  $\sin A + \cos A =\sqrt{2} \cos A$

⇒ $\sin A = (\sqrt{2} - 1 ) \cos A$

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

{By rationalization}

⇒ $(\sqrt{2} + 1) \sin A = \cos A$

⇒ $\sqrt{2} \sin A + \sin A = \cos A$

⇒ $\cos A - \sin A = \sqrt{2} \sin A$ (Proved)