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

Solution:

$\frac{\sin x+1}{\cos x}=2\cos x (\cos x\ne 0)$

$\Rightarrow 2{\sin }^{2}x+\sin x-1=0$

$\Rightarrow (2\sin x-1)(\sin x+1)=0$, $\Rightarrow \sin x=\frac{1}{2}$

but $\sin x+1\ne 0 \because \cos x\ne 0)$

$\Rightarrow x=\frac{\pi }{6},\frac{5\pi }{6}$ in $[0,2\pi ]$