Question : If $\sin(\theta+30^{\circ})=\frac{3}{\sqrt{12}}$, then the value of $\cos^{2}\theta$ is:
Option 1: $\frac{1}{4}$
Option 2: $\frac{\sqrt{3}}{2}$
Option 3: $\frac{3}{4}$
Option 4: $\frac{1}{2}$
Correct Answer: $\frac{3}{4}$
Solution :
Given: $\sin(\theta+30^{\circ})=\frac{3}{\sqrt{12}}$
⇒ $\sin(\theta+30^{\circ})=\frac{\sqrt{3}}{2}$
⇒ $\sin(\theta+30^{\circ})=\sin(60^{\circ})$
⇒ $\theta+30^{\circ}=60^{\circ}$
⇒ $\theta=30^{\circ}$
Substituting $\theta=30^{\circ}$, we get,
$\cos^{2}\theta=\cos^{2}(30^{\circ})=(\frac{\sqrt3}{2})^{2}=\frac{3}{4}$
Hence, the correct answer is $\frac{3}{4}$.
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