Question

Number of solutions of the equation $\tan x+\sec x=$ $2 \cos x$ lying in the interval $[0, \pi]$ is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (C)

Given, equation is $\tan x+\sec x=2 \cos x$
$\Rightarrow \frac{\sin x}{\cos x}+\frac{1}{\cos x}=2 \cos x $
$\Rightarrow 1+\sin x=2 \cos ^{2} x$
$\Rightarrow 1+\sin x=2\left(1-\sin ^{2} x \right)$
$\Rightarrow 2 \sin ^{2} x+2 \sin x-\sin x-1=0$
$\Rightarrow 2 \sin x(\sin x+1)-1(\sin x+1)=0$
$\Rightarrow (2 \sin x-1)(\sin x+1)=0$
$\Rightarrow $ either $\sin x=\frac{1}{2}$ or $\sin x=-1$
$\Rightarrow $ either $x=\frac{\pi}{6}, \frac{5 \pi}{6} \in[0, \pi]$ or $x=\frac{3 \pi}{2}$
But, $x=\frac{3 \pi}{2}$ can not be possible.
$\therefore $ Number of solutions are $2 .$