Question

Can some one help explain how to evaluate the following without a calculator?

sin^2(25) sin^2(65)

Answer 1

We know:
$\sin(90^o-\alpha)=\cos\alpha$

$\sin25^o=\sin(90^o-65^o)=\cos65^o\ therefore\ \sin^225^o=\cos^265^o$

We have:
$\sin^225^o+\sin^265^o=\cos^265^o+\sin^265^o=1$

Used: $\sin^2\alpha+\cos^2\alpha=1$