Question:
If $x \tan 45^{\circ} \cos 60^{\circ}=\sin 60^{\circ} \cot 60^{\circ}$, then $x$ is equal to
(a) 1
(b) $\sqrt{3}$
(c) $\frac{1}{2}$
(d) $\frac{1}{\sqrt{2}}$
Solution:
Given that: $x \tan 45^{\circ} \cos 60^{\circ}=\sin 60^{\circ} \cot 60^{\circ}$
Here we have to find the value of $x$
We know that $\left[\begin{array}{l}\tan 45^{\circ}=1 \\ \cos 60^{\circ}=\frac{1}{2} \\ \sin 60^{\circ}=\frac{\sqrt{3}}{2} \\ \cot 60^{\circ}=\frac{1}{\sqrt{3}}\end{array}\right]$
$\Rightarrow x \tan 45^{\circ} \cos 60^{\circ}=\sin 60^{\circ} \cot 60^{\circ}$
$\Rightarrow x \times 1 \times \frac{1}{2}=\frac{\sqrt{3}}{2} \times \frac{1}{\sqrt{3}}$
$\Rightarrow x=1$
Hence the correct option is $(a)$