# Aptitude Question ID : 94464

$\sin ^{2}5^{\circ}+\sin ^{2}10^{\circ}+\sin ^{2}15^{\circ}+......+\sin ^{2}85^{\circ}+\sin ^{2}90^{\circ}$ is equal to :
[A]$7\frac{1}{2}$
[B]$8\frac{1}{2}$
[C]$9\frac{1}{2}$
[D]$9$

$\mathbf{9\frac{1}{2}}$
$\sin \Theta = \cos\Theta (90^{\circ}-\Theta);$
$\sin (90^{\circ}-\Theta) = \cos\Theta$
$\therefore \sin 85^{\circ} = \sin (90^{\circ}-5^{\circ}) = \cos5^{\circ}$
$\mathbf{\therefore (\sin^{2}5^{\circ}+\sin^{2}85^{\circ})+(\sin^{2}10^{\circ}+\sin^{2}80^{\circ})+.... to 8 terms +(\sin^{2}45^{\circ}+\sin^{2}90^{\circ})}$
$= 8\times 1 + \frac{1}{2}+1 = 9\frac{1}{2}$
Hence option [C] is correct answer.