Question

If $\frac{sinx}{a}=\frac{cosx}{b}$,Prove that $asin2x+bcos2x=b.$

Answer 1

Given:- $\frac{\sin x}{a}=\frac{\cos x}{b}\Rightarrow b\sin x=a\cos x$

To prove:- $a\sin 2x+b\cos 2x=b$

Proof:-

Using trigonometric identity:

$\sin 2x=2\sin x\cos x$

$\cos 2x=1-2{\sin }^{2}x$

Therefore,

Taking L.H.S.-

$a\sin 2x+b\cos 2x$

$=a\left(2\sin x\cos x\right)+b(1-2{\sin }^{2}x)$

$=2\sin x\left(a\cos x\right)+b-2b{\sin }^{2}x$

$=2\sin x\left(b\sin x\right)+b-2b{\sin }^{2}x(\because a\cos x=b\sin x)$

$=2b{\sin }^{2}x+b-2b{\sin }^{2}x$

$=b$

$=$R.H.S.

Hence proved.