Prove the following identity 1-2sin^2a=cos^4a-sin^4a.

1 - 2sin62 a = cos^4 a - sin^4
a

First we will start from the right side and prove the
left sides.

==> cos^4 a - sin^4
a

Let us factor.

We know
that:

cos^4 a - sin^4 a = (cos^2 a - sin^2a)(cos^2 a +
sin^2 a)

But sin^2 a + cos^2 a =
1

==> cos^4 a - sin^2 a = (cos^2 a - sin^2
a)

But we know that:

cos^2 a +
sin^2 a = 1

==> sin^2 a = 1- cos^2
a

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

                                      = cos^2 a -1 +
cos^2 a

                                   = 2cos^2 a
-1

==> cos^4 a - sin^4 a = 2cos^2 a -1
...........q.e.d