Question

How do you prove `cos^4(x)-sin^4(x)=cos2x`?

Answer 1

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

Remember the double angle formula for cosine:
`color(blue)(cos(2x)=cos^2(x)-sin^2(x)`

Plugging it into the right hand side:
`cos^4(x) - sin^4(x) = cos^2(x)-sin^2(x)`

Using differences of squares on the left side:
`(cos^2(x) + sin^2(x)) (cos^2(x) - sin^2(x)) = cos^2(x)-sin^2(x)`

And since `color(blue)(cos^2(x) + sin^2(x)=1`
`1(cos^2(x) - sin^2(x)) = cos^2(x)-sin^2(x)`