The value of sin^2 5^{circ} + sin^2 10^{circ} | vedclass

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Question

(A) The given expression is $S = \sin^2 5^{\circ} + \sin^2 10^{\circ} + \ldots + \sin^2 85^{\circ} + \sin^2 90^{\circ}$.We know that $\sin^2 	heta +

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$\frac{19}{2}$

Vedclass · Answer

(A) The given expression is $S = \sin^2 5^{\circ} + \sin^2 10^{\circ} + \ldots + \sin^2 85^{\circ} + \sin^2 90^{\circ}$.We know that $\sin^2 	heta + \sin^2(90^{\circ} - 	heta) = \sin^2 	heta + \cos^2 	heta = 1$.There are $17$ terms from $5^{\circ}$ to $85^{\circ}$ in steps of $5^{\circ}$.These can be paired as $(\sin^2 5^{\circ} + \sin^2 85^{\circ}) + (\sin^2 10^{\circ} + \sin^2 80^{\circ}) + \ldots + (\sin^2 40^{\circ} + \sin^2 50^{\circ}) + \sin^2 45^{\circ}$.There are $8$ such pairs,each summing to $1$,and one middle term $\sin^2 45^{\circ} = (\frac{1}{\sqrt{2}})^2 = \frac{1}{2}$.So,the sum of these $17$ terms is $8 	imes 1 + \frac{1}{2} = 8.5$.Adding the last term $\sin^2 90^{\circ} = 1$,the total sum is $8.5 + 1 = 9.5 = \frac{19}{2}$.

The value of $\sin^2 5^{\circ} + \sin^2 10^{\circ} + \sin^2 15^{\circ} + \ldots + \sin^2 85^{\circ} + \sin^2 90^{\circ} = $

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