Question

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

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

$\tan x+\sec x=2\cos x$
$\Rightarrow \frac{\sin x}{\cos x}+\frac{1}{\cos x}=2\cos x$
$\Rightarrow \sin x+1=2{\cos }^{2}x$ for $\cos x\ne 0$
$\Rightarrow \sin x+1=2-2{\sin }^{2}x\Rightarrow 2{\sin }^{2}x+\sin x-1=0\Rightarrow (2\sin x-1)(\sin x+1=0)$
$\Rightarrow \sin x=\frac{1}{2}$
or $\sin x=-1$ (not possible as then $\cos x=0$ and $\sec x,$ $\tan x$ are not defined.)
$x=\frac{\pi }{6},\frac{5\pi }{6}$
Hence, number of solutions $=2$