Answer
The expression on the left-side is equal to the expression on the right-side.
Work Step by Step
The given expression on the left side $\tan \frac{x}{2}$ can be simplified by using the half angle formula $\tan \frac{x}{2}=\frac{1-\cos x}{\sin x}$.
$\begin{align}
& \tan \frac{x}{2}=\frac{1-\cos x}{\sin x} \\
& =\frac{1-\cos x}{\sin x}.\frac{\frac{1}{\cos x}}{\frac{1}{\cos x}} \\
& =\frac{\frac{1-\cos x}{\cos x}}{\frac{\sin x}{\cos x}}
\end{align}$
Further solving and using the quotient identity $\frac{\sin x}{\cos x}=\tan x$, gives:
$\begin{align}
& \frac{\frac{1-\cos x}{\cos x}}{\frac{\sin x}{\cos x}}=\frac{\frac{1}{\cos x}-\frac{\cos x}{\cos x}}{\tan x} \\
& =\frac{\sec x-1}{\tan x}
\end{align}$
Hence, the expression on the left-side is equal to the expression on the right-side.
