Question

How do you prove `cos^4x - sin^4x = 1 - 2sin^2x`?

Answer 1

Start by factoring the left side as a difference of squares.

Explanation:

`cos^4x - sin^4x = 1 - 2sin^2x`

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

Now, applying the pythagorean identity `cos^2x + sin^2x = 1`:

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

Rearranging the previously stated pythagorean identity to solve for sin:

`cos^2x + sin^2x = 1`

`cos^2x = 1 - sin^2x`

Substituting:

`1(1 - sin^2x - sin^2x) =`

`1 - 2sin^2x = 1 - 2sin^2x ->` identity proved

Hopefully this helps!