verify if cos^4a - sin^4a = cos2a

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.