The value of $\sin^{2} 5^{\circ} + \sin^{2} 10^{\circ} + \sin^{2} 15^{\circ} . . . . . . + \sin^{2} 90^{\circ}$ is equal to

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\frac{17}{2}

\frac{19}{2}

\frac{21}{2}

\frac{23}{2}

answer is B.

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Detailed Solution

Given expression is

\sin^{2} 5^{\circ} + \sin^{2} 10^{\circ} + \sin^{2} 15^{\circ} . . . . . . + \sin^{2} 90^{\circ}

.
We need to find the value of the given expression.
By arranging the terms, we get

= (\sin^{2} 5^{o} + \sin^{2} 85^{o}) + (\sin^{2} 10^{o} + \sin^{2} 80^{o}) + . . . . . . . . . . . + (\sin^{2} 45^{o} + \sin^{2} 45^{o}) + \sin^{2} 90^{o} - \sin^{2} 45^{o}

= (\sin^{2} 5^{o} + \cos^{2} 5^{o}) + (\sin^{2} 10^{o} + \cos^{2} 10^{o}) + . . . . . . . . . . . + (\sin^{2} 45^{o} + \cos^{2} 45^{o}) + \sin^{2} 90^{o} - \sin^{2} 45^{o}

= 1 + 1 + 1 + . . . . . . . + 1 - \frac{1}{2}

= 10 - \frac{1}{2}

= \frac{19}{2}

Hence, the correct option is (2).

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