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  4. The value of sin 2 0°+ sin 2 5°+ sin 2 10°+ ldots+ sin 2 85° is

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Q. The value of $\sin ^2 0^{\circ}+\sin ^2 5^{\circ}+\sin ^2 10^{\circ}+\ldots+$ $\sin ^2 85^{\circ}$ is ___

Trigonometry

Solution:

$\sin ^2 0^{\circ}+\sin ^2 5^{\circ}+\sin ^2 10^{\circ}+\ldots+\sin ^2 85^{\circ}$
$=\left(\sin ^2 5^{\circ}+\sin ^2 85^{\circ}\right)+\left(\sin ^2 10^{\circ}+\sin ^2 80^{\circ}\right) \ldots+\sin ^2 0^{\circ}+\sin ^2 45^{\circ} $
$\left(\sin 85^{\circ}=\cos 5^{\circ}\right)=8+0+\left(\frac{1}{\sqrt{2}}\right)^2=\frac{17}{2}=8.5$