# How do you use the trig identity cos2x = cos2 x - sin2 x to verify that cos 2x = 2 cos2 x -1?

We have to prove

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

### Proof

$\cos 2x =\cos ^{2}x -\sin ^{2}x$——(i)

We have learnt that $\cos ^{2}x + \sin ^{2}x=1$

So $\sin ^{2}x=1 – \cos ^{2}x$

Substituting in equation (i) we get

$\cos 2x =\cos ^{2}x -(1-\cos ^{2}x)$

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

Hence Proved