Prove algebraically that the equation 1 2 cos x1 2 cos x

Prove algebraically that the equation (1 + 2 cos x)(1 - 2 cos x) = 4 sin² x - 3 is an identity.

Solution

(1 + 2 cos x)(1 - 2 cos x) = 4 sin2 x - 3

LHS :(1 + 2 cos x)(1 - 2 cos x) = 1 -4cos^2x ( expand using (a+b)(a-b) =a^2 -b^2 )

=1 -4cos^2x

We can use Trig identity : sin^2x +cos^2x =1

= 1 -4 ( 1-sin^2x) = 1- 4 +4sin^2x

= 4sin^2x -3

= RHS