Question

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

Answer 1

The given equation is $\tan x+\sec x=2\cos x$
$\Rightarrow \frac{\sin x}{\cos x}+\frac{1}{\cos x}=2\cos x$
$\Rightarrow \sin x+1=2{\cos }^{2}x=2-2{\sin }^{2}x$
$\Rightarrow 2{\sin }^{2}x+\sin x-1=0$
$\Rightarrow (2\sin x-1)(\sin x+1)=0$
$\Rightarrow \sin x=-1,\frac{1}{2}$
$\Rightarrow x=\frac{\pi }{6},\frac{5\pi }{6},\frac{3\pi }{2}\in [0,2\pi ]$
But for $x=3\pi /2,\tan x$ and $\sec x$ are not defined.
Therefore, there are only two solutions.