Question

Evaluate the following and justify your answer :
$\frac{{\sin }^2{15}^{\circ }+{\sin }^2{75}^{\circ }}{{\cos }^2{36}^{\circ }+{\cos }^2{54}^{\circ }}$

Answer 1

$\frac{{\sin }^2{15}^{\circ }+{\sin }^2{75}^{\circ }}{{\cos }^2{36}^{\circ }+{\cos }^2{54}^{\circ }}$

$=\frac{{\sin }^2{15}^{\circ }+{\sin }^2({90}^{\circ }-{15}^{\circ })}{{\cos }^2{36}^{\circ }+{\cos }^2({90}^{\circ }-{36}^{\circ })}$

$=\frac{{\sin }^2{15}^{\circ }+{\cos }^2{15}^{\circ }}{{\cos }^2{36}^{\circ }+{\sin }^2{36}^{\circ }}$

$=\frac{1}{1}=1$ since ${\sin }^2\theta +{\cos }^2\theta =1$