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

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