Question

The value of
${\sin }^2{5}^{\circ }+{\sin }^2{10}^{\circ }+{\sin }^2{15}^{\circ }+{\sin }^2{75}^{\circ }+{\sin }^2{80}^{\circ }+{\sin }^2{85}^{\circ }+{\sin }^2{90}^{\circ }\text{is}$

Answer 1

We know that
${\sin }^2{85}^{\circ }={\sin }^2({90}^{\circ }-5^{\circ })={\cos }^2{5}^{\circ }$

$\therefore {\sin }^2{5}^{\circ }+{\sin }^2{85}^{\circ }={\sin }^2{5}^{\circ }+{\cos }^2{5}^{\circ }=1$

Similarly ${\sin }^2{10}^{\circ }+{\sin }^2{80}^{\circ }={\sin }^2{15}^{\circ }+{\sin }^2{75}^{\circ }=1$

Hence ${\sin }^2{5}^{\circ }+{\sin }^2{10}^{\circ }+{\sin }^2{15}^{\circ }+{\sin }^2{75}^{\circ }+{\sin }^2{80}^{\circ }+{\sin }^2{85}^{\circ }+{\sin }^2{90}^{\circ }=1+1+1+1=4$