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.**

We'll write the difference of squares (cos a)^4 - (sin a)^4 using the formula:

x^2 - y^2 = (x-y)(x+y)

We'll put x = (cos a)^2 and y = (sin a)^2

(cos a)^4 - (sin a)^4 = [(cos a)^2 - (sin a)^2][(cos a)^2 + (sin a)^2] (1)

We'll write cos 2a = cos (a+a)

cos (a+a) = cosa*cosa - sina*sina

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

We'll substitute (1) and (2) in the given expression:

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

We'll divide by (cos a)^2 - (sin a)^2:

**(cos a)^2 + (sin a)^2 = 1 true!**

**The relation above is the fundamental formula of trigonometry.**