We have to prove: sin^4(theta) - cos^4(theta) = sin^2(theta) - cos^2(theta)

First let's write the terms in a standard form and use x instead of theta.

So we have to prove (sin x)^4 - (cos x)^4 = (sin x)^2 - (cos x)^2

Start with the left hand side:

(sin x)^4 - (cos x)^4

we use the relation x^2 - y^2 = (x - y)(x + y)

=> [(sin x)^2 - (cos x)^2][(sin x)^2 + (cos x)^2]

we know that [(sin x)^2 + (cos x)^2] = 1

=> [(sin x)^2 - (cos x)^2] * 1

=> [(sin x)^2 - (cos x)^2]

which is the right hand side.

This proves that sin^4(theta) - cos^4(theta) = sin^2(theta) - cos^2(theta)