Verify the following identity by using an angle sum identity

Verify the following identity by using an angle sum identity: cos (2x) = 1 – 2(sin2 x). Hint (2x = x + x)

Solution

We know that cos(x + y) = cosx cos y – sin x sin y.

Therefore, cos (2x ) = cos (x + x) = cosx cosx – sinx sinx = cos² x – sin² x = (1 –sin² x ) - sin² x = 1 -2 sin² x ( as sin² x + cos² x = 1)