Question : If $k\left(\tan 45^{\circ} \sin 60^{\circ}\right)=\cos 60^{\circ} \cot 30^{\circ}$, then the value of k is:
Option 1: $1$
Option 2: $2$
Option 3: $\frac{1}{\sqrt{3}}$
Option 4: $\sqrt{3}$
Correct Answer: $1$
Solution :
The given equation is $k\left(\tan 45^{\circ} \sin 60^{\circ}\right)=\cos 60^{\circ} \cot 30^{\circ}$.
$⇒k\left(1 \times \frac{\sqrt{3}}{2}\right) = \frac{1}{2} \times \sqrt{3}$
$⇒k = \frac{\frac{1}{2} \times \sqrt{3}}{\frac{\sqrt{3}}{2}} = \frac{1}{2} \times \frac{2}{\sqrt{3}} \times \sqrt{3} = 1$
Hence, the correct answer is $1$.
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