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 $2.$