Answer
The value of $x$ is $\frac{\pi }{3},\frac{5\pi }{3},\pi $.
Work Step by Step
The trigonometric equation is
$\begin{align}
& {{\tan }^{2}}x-\sec x-1=0 \\
& \left( {{\sec }^{2}}x-1 \right)-\sec x-1=0 \\
& {{\sec }^{2}}x-\sec x-2=0 \\
& {{\sec }^{2}}x-2\sec x+\sec x-2=0
\end{align}$
And,
$\begin{align}
\left( \sec x-2 \right)\left( \sec x+1 \right)=0 & \\
\left( \sec x-2 \right)=0\text{ or }\left( \sec x+1 \right)=0 & \\
\sec x=2,-1 & \\
x=\frac{\pi }{3},\frac{5\pi }{3},\pi & \\
\end{align}$
