Prove that

To Prove: $\cos 2 x+2 \sin ^{2} x=1$

Taking LHS,

$=\cos 2 x+2 \sin ^{2} x$

$=\left(2 \cos ^{2} x-1\right)+2 \sin ^{2} x\left[\because 1+\cos 2 x=2 \cos ^{2} x\right]$

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

$=2(1)-1\left[\because \cos ^{2} \theta+\sin ^{2} \theta=1\right]$

$=2-1$

$=1$

= RHS

∴ LHS = RHS

Hence Proved