Question

The smallest value of $x$ satisfying the equation $\sqrt{3}(\cot x+\tan x)=4\text{is}$

• A. $2\pi /3$
• B. $\pi /3$
• C. $\pi /6$
• D. $\pi /12$

Answer 1

The correct option is C $\pi /6$

Given : $\sqrt{3}(\cot x+\tan x)=4$

Substituting $\cot x=\frac{\cos x}{\sin x}$ and $\tan x=\frac{\sin x}{\cos x},$

we get :

$=\sqrt{3}(\frac{\cos x}{\sin x}+\frac{\sin x}{\cos x})=4$

$=\sqrt{3}\left(\frac{{\cos }^{2}x+{\sin }^{2}x}{\sin x\cos x}\right)=4$

Using identity ${\cos }^{2}x+{\sin }^{2}x=1$ and $2\sin x\cos x=\sin 2x,$ we get :

$\Rightarrow \sqrt{3}\left(1\right)=4\sin x\cos x$

$\Rightarrow \sqrt{3}=2\left(2\sin x\cos x\right)$

$\Rightarrow \frac{\sqrt{3}}{2}=\sin 2x$

$\Rightarrow \sin \frac{\pi }{3}=\sin 2x$

$\Rightarrow 2x=\frac{\pi }{3}$

$\Rightarrow x=\frac{\pi }{6}$