cos^4 a - sin^4 a = cos2a

Let us rewrite:

(cos^2 a)^2 - (sin^2 a)^2

We kno wthat:

a^2 - b^2 = (a-b)(a+b)

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

Also:

We know that:

cos^2 a-sin^2 a = cos2a

cos^2 a + sin^2 a=...

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cos^4 a - sin^4 a = cos2a

Let us rewrite:

(cos^2 a)^2 - (sin^2 a)^2

We kno wthat:

a^2 - b^2 = (a-b)(a+b)

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

Also:

We know that:

cos^2 a-sin^2 a = cos2a

cos^2 a + sin^2 a= 1

Now substitute:

cos^4 a - sin^4 a= cos2a * 1

= cos2a

**==> the equality is true.**