Q169 / 270 · NDA 2015 (Phase 2) · Mathematics
The value of $\sin ^2 5^{\circ}+\sin ^2 10^{\circ}+\sin ^2 15^{\circ}+\sin ^2 20^{\circ}+\ldots$. $+\sin ^2 90^{\circ}$ is
- A7
- B8
- C9
- Correct answerD19/2
Solution
$\begin{aligned}
& \sin ^2 5^{\circ}+\sin ^2 10^{\circ}+\sin ^2 15^{\circ}+\ldots \ldots \ldots . .+\ldots \ldots \ldots . .+\sin ^2 75^{\circ}+ \\
& \sin ^2 80^{\circ}+\sin ^2 85^{\circ}+\sin ^2 90^{\circ} \\
& \Rightarrow \sin ^2 5^{\circ}+\sin ^2 10^{\circ}+\sin ^2 15^{\circ}+\ldots \ldots . . \sin ^2\left(90-15^{\circ}\right)+ \\
& \sin ^2(90-10)+\sin ^2(90-5)+1 \\
& \Rightarrow \sin ^2 5+\sin ^2 10+\sin ^2 15^{\circ}+\ldots \ldots . .+\cos ^2 15^{\circ}+ \\
& \cos ^2 10^{\circ}+\cos ^2 5^{\circ}+1 \\
& \Rightarrow(1+1+1+\ldots .8 \text { times })+\sin ^2 45^{\circ}+1 \\
& \Rightarrow 8+\frac{1}{2}+1=\frac{19}{2}
\end{aligned}$
Hence, the correct answer is option: 'D'.