Without using tables, evaluate the following: ^230^ ^245^ + ^260^… - Vidyadip

Question

Without using tables, evaluate the following: $\sin^230^\circ \sin^245^\circ + \sin^260^\circ \sin^290^\circ$.

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Answer

$\sin ^2 30^{\circ} \sin ^2 45^{\circ}+\sin ^2 60^{\circ} \sin ^2 90^{\circ}$
$\sin 30^{\circ}=\frac{1}{2}$
$\sin 45^{\circ}=\frac{1}{\sqrt{2}}$
$\sin 60^{\circ}=\frac{\sqrt{3}}{2}$
$\sin 90^{\circ}=1$
$\sin ^2 30^{\circ} \sin ^2 45^{\circ}+\sin ^2 60^{\circ} \sin ^2 90^{\circ}$
$=\left(\frac{1}{2}\right)^2\left(\frac{1}{\sqrt{2}}\right)^2+\left(\frac{\sqrt{3}}{2}\right)^2 1$
$=\frac{1}{4} \times \frac{1}{2}+\frac{3}{4}$
$=\frac{1}{8}+\frac{3}{4}$
$=\frac{1+6}{8}$
$=\frac{7}{8} .$

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