Question : What is the value of $\sin 75^{\circ}+\sin 15^{\circ}$?
Option 1: $\frac{1}{\sqrt{2}}$
Option 2: $\frac{\sqrt{3}}{2}$
Option 3: $\sqrt{\frac{3}{2}}$
Option 4: $\frac{3}{\sqrt{2}}$
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Correct Answer: $\sqrt{\frac{3}{2}}$
Solution :
Let $t=\sin 15^\circ+\sin75^\circ$
Now, $t=\sin 15^\circ+\cos (90-75)^\circ$
$⇒t=\sin 15^\circ+\cos 15^\circ$
Squaring on both sides, we get,
$t^2=\sin^2 15^\circ+\cos^2 15^\circ+2\sin15^\circ \cos15^\circ$
$⇒t^2=1+\sin 30^\circ$
$⇒t^2=1+\frac{1}{2}$
$\therefore t=\sqrt{\frac{3}{2}}$
Hence, the correct answer is $\sqrt{\frac{3}{2}}$.
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