Answer
The solutions for the equation are $x=\frac{\pi }{2},\frac{3\pi }{2},\frac{\pi }{3},\frac{4\pi }{4},\frac{2\pi }{3},\frac{5\pi }{3}$.
Work Step by Step
$\begin{align}
& \cos x\cdot {{\tan }^{2}}x=3\cos x \\
& \cos x\cdot {{\tan }^{2}}x-3\cos =0 \\
& \cos x\left( {{\tan }^{2}}x-3 \right)=0
\end{align}$
If $\cos x=0$
Then,
$x=\frac{\pi }{2},\frac{3\pi }{2}$
$\begin{align}
& {{\tan }^{2}}x-3=0 \\
& {{\tan }^{2}}x=3
\end{align}$
$\begin{align}
& \tan x=\pm \sqrt{3} \\
& \tan x=\sqrt{3} \\
\end{align}$
$x=\frac{\pi }{3},\frac{4\pi }{4}$
$\tan x=-\sqrt{3}$
$\therefore x=\frac{2\pi }{3},\frac{5\pi }{3}$
Hence, the solutions for the equation are $x=\frac{\pi }{2},\frac{3\pi }{2},\frac{\pi }{3},\frac{4\pi }{4},\frac{2\pi }{3},\frac{5\pi }{3}$.
