The value of ${sin}^{2} 5^{o} + {sin}^{2} 10^{o} + {sin}^{2} 15^{o} + {sin}^{2} 20^{o} + . . . . . . . . . . . . . + {sin}^{2} 90^{o}$ is

7

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8

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9

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\frac{19}{2}

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Solution

The correct option is D $\frac{19}{2}$
Solution:
${sin}^{2} 5^{\circ} + {sin}^{2} 10^{\circ} + {sin}^{2} 15^{\circ} + {sin}^{2} 20^{\circ} + . . . . . . . . . . . . . . . . . . . . + {sin}^{2} 90^{\circ}$
$= ({sin}^{2} 5^{\circ} + {sin}^{2} 85^{\circ}) + ({sin}^{2} 10^{\circ} + {sin}^{2} 80^{\circ}) + . . . . . . . . . . . . . . . + ({sin}^{2} 40^{\circ} + {sin}^{2} 50^{\circ}) + {sin}^{2} 45^{\circ} + {sin}^{2} 90^{\circ}$
$= ({sin}^{2} 5^{\circ} + {cos}^{2} 5^{\circ}) + ({sin}^{2} 10^{\circ} + {cos}^{2} 10^{\circ}) + . . . . . . . . . . . . . . . + ({sin}^{2} 40^{\circ} + {cos}^{2} 40^{\circ}) + {sin}^{2} 45^{\circ} + {sin}^{2} 90^{\circ}$
$= 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + \frac{1}{2} + 1$
$= \frac{19}{2}$
Hence, D is the correct option.

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