# How do you prove 1 - 2 sin^2r + sin^4r = cos^4r?

Use the fact that $1 + 2 {x}^{2} + {x}^{4}$ can be factored. It is a perfect square.
$1 + 2 {x}^{2} + {x}^{4} = {\left(1 - {x}^{2}\right)}^{2}$.
$1 - 2 {\sin}^{2} r + {\sin}^{4} r = {\left(1 - {\sin}^{2} r\right)}^{2} = {\left({\cos}^{2} r\right)}^{2} = \cos 4 r$