How do you verify the identity sin^2alpha-sin^4alpha=cos^2alpha-cos^4alpha?

Feb 26, 2017

$\text{please look at the fallowing solution.}$

Explanation:

${\sin}^{2} \alpha - {\sin}^{4} \alpha = \textcolor{red}{{\cos}^{2} \alpha - {\cos}^{4} \alpha}$

$\text{let } {\sin}^{2} \alpha - {\sin}^{4} \alpha = k$

${\sin}^{2} \alpha = 1 - {\cos}^{2} \alpha$

${\sin}^{4} \alpha = {\left(1 - {\cos}^{2} \alpha\right)}^{2} = 1 - 2 {\cos}^{2} \alpha + {\cos}^{4} \alpha$

$k = 1 - {\cos}^{2} \alpha - \left(1 - 2 {\cos}^{2} \alpha + {\cos}^{4} \alpha\right)$

$k = \cancel{1} - \cancel{{\cos}^{2}} \alpha \cancel{- 1} + \cancel{2 {\cos}^{2}} \alpha - {\cos}^{4} \alpha$

$k = \textcolor{red}{{\cos}^{2} \alpha - {\cos}^{4} \alpha}$