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
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