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

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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

 

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We have to prove that  1 - 2(sin a)^2 = (cos a)^4  - (sin a)^4.

We know that (sin a)^2 + (cos a)^2 = 1

(cos a)^4  - (sin a)^4

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

=> [(cos a)^2  - (sin a)^2]

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

=1- 2 (sin a )^2

Therefore 1 - 2(sin a)^2 = (cos a)^4  - (sin a)^4.

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