Question

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

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (C)

$\tan x+\sec x=2 \cos x$
$\Rightarrow \frac{\sin x}{\cos x}+\frac{1}{\cos x}=2 \cos x$
$\Rightarrow \sin x+1=2 \cos ^{2} x=2\left(1-\sin ^{2} x\right)$
$=2-2 \sin ^{2} x$
$\Rightarrow 2 \sin ^{2} x+\sin x-1=0$
$\Rightarrow 2 \sin ^{2} x+2 \sin x-\sin x-1=0$
$\Rightarrow 2 \sin x(\sin x+2)-1(\sin x+1)=0$
$\Rightarrow (\sin x+2)(2 \sin x-1)=0$
$\Rightarrow \sin x=-2$ which is not possible.
or $\sin x=\frac{1}{2}$
$\Rightarrow x=\frac{\pi}{6}, \frac{5 \pi}{6}$
$x \in[0, \pi]$
$\therefore $ The number of solution $=2$