Question

The smallest value of x satisfying the equation

\[\sqrt{3} \left( \cot x + \tan x \right) = 4\] is 

Options

• \[2\pi/3\]

   

• `pi/3`

• `pi/6`

• `pi/12`

Answer 1

\[\pi/6\]
Given:

\[\sqrt{3}(\cot x + \tan x) = 4\]

\[ \Rightarrow \sqrt{3} \left( \frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} \right) = 4\]

\[ \Rightarrow \sqrt{3} ( \cos^2 x + \sin^2 x) = 4 \sin x \cos x\]

\[ \Rightarrow \sqrt{3} = 2 \sin2x [\sin2x = 2 \sin x \cos x]\]

\[ \Rightarrow \sin2x = \frac{\sqrt{3}}{2}\]

\[ \Rightarrow \sin2x = \sin \frac{\pi}{3}\]

\[ \Rightarrow 2x = n\pi + ( - 1 )^n \frac{\pi}{3}, n \in Z\]

\[ \Rightarrow x = \frac{n\pi}{2} + ( - 1 )^n \frac{\pi}{6}, n \in Z\]
To obtain the smallest value of x, we will put n = 0 in the above equation.
Thus, we have:
`x=pi/6`
Hence, the smallest value of x is 
`pi/6`.