Question : If $\sin (\theta +18^{\circ})=\cos 60^{\circ}(0< \theta < 90^{\circ})$, then the value of $\cos 5\theta$ is:
Option 1: $\frac{1}{2}$
Option 2: $0$
Option 3: $\frac{1}{\sqrt{2}}$
Option 4: $1$
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Correct Answer: $\frac{1}{2}$
Solution :
$\sin (\theta +18^{\circ})=\cos 60^{\circ}$
We know that $\cos (90^{\circ}-\theta)=\sin\theta$
$⇒\sin (\theta +18^{\circ})=\cos(90^{\circ}-30^{\circ})$
$⇒\sin (\theta +18^{\circ})=\sin30^{\circ}$
$⇒\theta +18^{\circ}=30^{\circ}$
$⇒\theta=12^{\circ}$
$\therefore \cos 5\theta=\cos 5×12^{\circ}=\cos 60^{\circ}=\frac{1}{2}$
Hence, the correct answer is $\frac{1}{2}$.
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