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

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(D) The given expression is $S = \sin^2 5^\circ + \sin^2 10^\circ + \dots + \sin^2 85^\circ + \sin^2 90^\circ$.This series contains $18$ terms (from $

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$9\frac{1}{2}$

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(D) The given expression is $S = \sin^2 5^\circ + \sin^2 10^\circ + \dots + \sin^2 85^\circ + \sin^2 90^\circ$.This series contains $18$ terms (from $5^\circ$ to $90^\circ$ in steps of $5^\circ$).We know that $\sin^2 	heta + \sin^2(90^\circ - 	heta) = \sin^2 	heta + \cos^2 	heta = 1$.Pairing the terms: $(\sin^2 5^\circ + \sin^2 85^\circ) + (\sin^2 10^\circ + \sin^2 80^\circ) + \dots + (\sin^2 40^\circ + \sin^2 50^\circ) + \sin^2 45^\circ + \sin^2 90^\circ$.There are $8$ such pairs,each summing to $1$.Thus,$S = 8(1) + \sin^2 45^\circ + \sin^2 90^\circ$.Since $\sin 45^\circ = \frac{1}{\sqrt{2}}$,$\sin^2 45^\circ = \frac{1}{2}$.Since $\sin 90^\circ = 1$,$\sin^2 90^\circ = 1$.Therefore,$S = 8 + \frac{1}{2} + 1 = 9\frac{1}{2}$.

The value of $\sin^2 5^\circ + \sin^2 10^\circ + \sin^2 15^\circ + \dots + \sin^2 85^\circ + \sin^2 90^\circ$ is equal to

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