Precalculus: Establish the identity

Question

sin^2x cos^2x =1/4 sin^2(2x)

Answer 1

Work on the right hand side picturing as 1/4(sin(2x))².

Replace sin(2x) with 2 sin(x) cos(x) --double angle identity.

That gives: 1/4 (2sinxcosx)²

Now apply the exponent to each factor inside the parentheses.

1/4(4 sin²(x)cos²(x))

Finally 1/4 (4) = 1, resulting in sin²(x)cos²(x)

QED.

Answer 2

Convert cos² x to 1 - sin² x, multiply and rearrange/