First slide

Trigonometric Identities

Question

The value of $\sin^{2} 5^{\circ} + \sin^{2} 10^{\circ} + \sin^{2} 15^{\circ} + \dots + \sin^{2} 85^{\circ} + \sin^{2} 90^{\circ}$ is

Easy

A

$7 \frac{1}{2}$
B

$8 \frac{1}{2}$
C

$9 \frac{1}{2}$
D

none of these

Solution

We have,
$\begin{array}{l} \sin^{2} 5^{\circ} + \sin^{2} 10^{\circ} + \sin^{2} 15^{\circ} + \dots . + \sin^{2} 85^{\circ} + \sin^{2} 90^{\circ} \\ = (\sin^{2} 5^{\circ} + \sin^{2} 85^{\circ}) + (\sin^{2} 10^{\circ} + \sin^{2} 80^{\circ}) + \dots \dots \\ \dots + \dots + (\sin^{2} 40^{\circ} + \sin^{2} 50^{\circ}) + (\sin^{2} 45^{\circ} + \sin^{2} 90^{\circ}) \\ = 8 + \frac{1}{2} + 1 = 9 \frac{1}{2} \end{array}$

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