Question

How do you verify `cos^4(x)-sin^4(x)=cos2x` using the double angle identity?

Answer 1

`cos(2x)=cos(x+x)=cos^2x-sin^2x`

Explanation:

Factor the left hand side. It is the difference of two squares.

`(cos^2x+sin^2x)(cos^2x-sin^2x)=cos(2x)`

From the Pythagorean identity we know that

`cos^2x+sin^2x=1` so we can write

`cos^2x-sin^2x=cos(2x)`

Now

`cos(2x)=cos(x+x)`

From the angle sum identity we have

`cos(2x)=cos(x+x)=cos(x)cos(x)-sin(x)sin(x)`

`cos(2x)=cos(x+x)=cos^2x-sin^2x`

Which shows that the original equation is true

Answer 2

Simplify `cos^4 x - sin^4 x`

Explanation:

`cos^4 x - sin^4 x = (cos^2 x + sin^2 x) (cos^2 x - sin^2 x)`=
= cos 2x

Reminder of 2 trig identities:
`cos^2 x + sin^2 x = 1`
`cos^2 x - sin^2 x = cos 2x`