The number of solutions of the equation $\tan x+\sec x=2\cos x$ lying in the interval $\left[\mathrm{0,5}\pi \right]$ is

$\tan x+\sec x=2\cos x$
$\Rightarrow \sin x+1=2{\cos }^{2}x=2-2{\sin }^{2}x$
$\Rightarrow 2{\sin }^{2}x+\sin x-1=0$
$\Rightarrow \sin x=\frac{1}{2},-1$
$\sin x=-1\Rightarrow \cos x=0$ and hence $\tan x, \sec x$ is not defined
So, $\sin x=\frac{1}{2}$ 

Hence, in $\left[\mathrm{0,5}\pi \right]$ there are $6$ solutions.