Question

\[ \text{If } \tan x+\sec x=\sqrt3,\ 0<x<\pi, \]

\[ \text{then } x \text{ is equal to} \]

(a) \(\frac{5\pi}{6}\)
(b) \(\frac{2\pi}{3}\)
(c) \(\frac{\pi}{6}\)
(d) \(\frac{\pi}{3}\)

Solution

Using identity,

\[ \sec x+\tan x = \tan\left(\frac{\pi}{4}+\frac{x}{2}\right) \]

So,

\[ \tan\left(\frac{\pi}{4}+\frac{x}{2}\right)=\sqrt3 \]

\[ \frac{\pi}{4}+\frac{x}{2}=\frac{\pi}{3} \]

\[ \frac{x}{2}=\frac{\pi}{3}-\frac{\pi}{4} \]

\[ \frac{x}{2}=\frac{\pi}{12} \]

\[ x=\frac{\pi}{6} \]

Answer

\[ \boxed{\frac{\pi}{6}} \]

Correct Option: (c)