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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
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$+\sin ^{2} 90^{\circ}$ is
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Verified Answer
The correct answer is:
$\frac{19}{2}$
$\sin ^{2} 5^{\circ}+\sin ^{2} 10^{\circ}+\sin ^{2} 15^{\circ}+\ldots \ldots \ldots .+\ldots \ldots \ldots .+\sin ^{2} 75^{\circ}+$
$\sin ^{2} 80^{\circ}+\sin ^{2} 85^{\circ}+\sin ^{2} 90^{\circ}$
$\Rightarrow \sin ^{2} 5^{\circ}+\sin ^{2} 10^{\circ}+\sin ^{2} 15^{\circ}+\ldots \ldots . \sin ^{2}\left(90-15^{\circ}\right)+$
$\sin ^{2}(90-10)+\sin ^{2}(90-5)+1$
$\Rightarrow \sin ^{2} 5+\sin ^{2} 10+\sin ^{2} 15^{\circ}+\ldots \ldots .+\cos ^{2} 15^{\circ}+$
$\cos ^{2} 10^{\circ}+\cos ^{2} 5^{\circ}+1$
$\Rightarrow(1+1+1+\ldots .8$ times $)+\sin ^{2} 45^{\circ}+1$
$\Rightarrow 8+\frac{1}{2}+1=\frac{19}{2}$
$\sin ^{2} 80^{\circ}+\sin ^{2} 85^{\circ}+\sin ^{2} 90^{\circ}$
$\Rightarrow \sin ^{2} 5^{\circ}+\sin ^{2} 10^{\circ}+\sin ^{2} 15^{\circ}+\ldots \ldots . \sin ^{2}\left(90-15^{\circ}\right)+$
$\sin ^{2}(90-10)+\sin ^{2}(90-5)+1$
$\Rightarrow \sin ^{2} 5+\sin ^{2} 10+\sin ^{2} 15^{\circ}+\ldots \ldots .+\cos ^{2} 15^{\circ}+$
$\cos ^{2} 10^{\circ}+\cos ^{2} 5^{\circ}+1$
$\Rightarrow(1+1+1+\ldots .8$ times $)+\sin ^{2} 45^{\circ}+1$
$\Rightarrow 8+\frac{1}{2}+1=\frac{19}{2}$
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