Given sin^4 x - cos^4 x = 2sin^2 x - 1

==> We will rearrange terms.

==> sin^4 x - 2sin^2 x + 1 = cos^4 x

Now we will factor the left side.

==> (sin^2 x -1)^2 = cos^ 4 x

But we know that sin62 x = 1- cos^2 x

==> (1-cos^2 x -1 )^2 = cos^4 x

==> (-cos^2 x)^2 = cos^4 x

==> cos^2 4 x = cos^4 x

Then, we have proved that sin^4 x - cos^4 x = 2sin^2 x - 1.....

We have to prove : sin^4x - cos^4x = 2sin^2x - 1

Let's start from the left

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

=> (sin x)^4 - [ 1 - (sin x)^2]^2

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

=> - 1 + 2*(sin x)^2

=> 2*(sin x)^2 - 1

which is the right hand side.

This proves that sin^4x - cos^4x = 2sin^2x - 1