Question

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

• A. $8$
• B. $9$
• C. $10$
• D. $9.5$

Answer 1

Answer: (D)

$\sin^{2} 5^{\circ} +\sin^{2} 10^{\circ } +\sin^{2} 15^{\circ } +... +\sin ^{2} 90^{\circ } $
$ = \sin ^{2} 5^{\circ } +\sin^{2} 10^{\circ }+....+\sin^{2} 35^{\circ } +\sin ^{2} 40^{\circ } + \sin ^{2} 45^{\circ } +\sin ^{2} \left(90^{\circ} - 40^{\circ}\right) +\sin ^{2} \left(90^{\circ } - 35^{\circ }\right)+....+\sin ^{2} \left(90^{\circ } - 15\right)+\sin ^{2} \left(90^{\circ } - 10^{\circ }\right) +\sin ^{2} \left(90^{\circ } - 5^{\circ }\right)+\sin ^{2} 90^{\circ } $
$ = \sin ^{2} 5^{\circ}+\sin ^{2}10^{\circ} + ....+\sin ^{2} 35^{\circ} +\sin ^{2}40^{\circ}+\sin ^{2} 45^{\circ}+\cos ^{2} 40^{\circ}+\cos ^{2} 35^{\circ}+....+ \cos ^{2} 10^{\circ} +\cos ^{2}5^{\circ} +\sin ^{2} 90^{\circ} $
$= \left[\left( \sin^{2} 5^{\circ} + \cos^{2} 5^{\circ} \right)+\left( \sin ^{2} 10^{\circ } + \cos ^{2} 10^{\circ } \right) + ......+ \sin ^{2} 40^{\circ } + \cos ^{2} 40^{\circ }\right]+\sin^{2} 45^{\circ} + \sin^2 90^{\circ} $
$ = [1 + 1 +.....+1 ] + \left( \frac{1}{\sqrt{2}}\right)^2 + 1 $
$= 8 + \frac{1}{2} + 1 = \frac{19}{2} = 9.5 $