Evaluate:
${sin}^{2} 5^{o} + {sin}^{2} 10^{o} + . . . . . . . . . . {sin}^{2} 85^{o}$ ?

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Solution

$= s i n^{2} 5 + s i n^{2} 10 + \dots \dots + s i n^{2} 75 + s i n^{2} 80 + s i n^{2} 85 +$

$= (s i n^{2} 5 + s i n^{2} 85) + (s i n^{2} 10 + s i n^{2} 80) + \dots \dots . . + s i n^{2} 44 + . . .$

$= (s i n^{2} 5 + c o s^{2} 5) + (s i n^{2} 10 + c o s^{2} 10) + \dots \dots . .$

$= (1 + 1 + 1 + 1 = . . .40 t i m e s) + \frac{1}{2}$

$= 40 + \frac{1}{2}$

$= 40.5$

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