Answer
The solutions to the given equation are $0,\,\frac{\pi }{3},\,\,\pi ,\ \text{ and }\ \frac{5\pi }{3}$.
Work Step by Step
We have to solve the equation on the interval $[0,2\pi )$:
$\begin{align}
& \text{tan }x\ \text{sec }x=2\text{tan }x \\
& \text{tan }x\ \text{sec }x-2\text{tan }x=0 \\
& \text{tan }x\left( \text{sec }x-2 \right)=0
\end{align}$
And each factor needs to be calculated as:
$\text{tan }x=0$
or
$\begin{align}
& \text{sec }x-2=0 \\
& \sec x=0+2 \\
& \sec x=2
\end{align}$
Then, solve for $x$ on the interval $[0,2\pi )$.
In the quadrant graph, the value of tangent is $0$ at $0$ and $\pi $. This implies,
$\begin{align}
& \text{tan }x=\tan 0 \\
& x=0
\end{align}$
$\begin{align}
& \tan x=\tan \pi \\
& x=\pi
\end{align}$
And the value of secant is not shown in the quadrant graph. By using the reciprocal identity of trigonometry, $\sec x=\frac{1}{\cos x}$ the secant function gets converted into the cosine function as,
$\begin{align}
& \frac{1}{cosx}=2 \\
& \cos x=\frac{1}{2}
\end{align}$
So, in the quadrant graph, the value of cosine is $\frac{1}{2}$ at $\frac{\pi }{3}$ and $\frac{5\pi }{3}$. This implies,
$\begin{align}
& \cos x=\cos \frac{\pi }{3} \\
& x=\frac{\pi }{3}
\end{align}$
$\begin{align}
& \cos x=\cos \frac{5\pi }{3} \\
& x=\frac{5\pi }{3}
\end{align}$
These are the proposed solutions of the tangent and cosine functions. Thus, the actual solutions in the interval $[0,2\pi )$ will be $0,\,\frac{\pi }{3},\,\,\pi ,\ \text{ and }\ \frac{5\pi }{3}$.
