Answer
The left side is equal to the right side.
Work Step by Step
We have the expression on the left side: $\sec x-\cos x$, which can be simplified by using the reciprocal identity $\sec x=\frac{1}{\cos x}$.
$\sec x-\cos x=\frac{1}{\cos x}-\cos x$
And the expression can be further simplified by multiplying and dividing the expression by $\cos x$.
$\begin{align}
& \sec x-\cos x=\frac{1}{\cos x}-\cos x \\
& =\frac{1}{\cos x}-\frac{\cos x}{1}.\frac{\cos x}{\cos x} \\
& =\frac{1-{{\cos }^{2}}x}{\cos x}
\end{align}$
We know the Pythagorean theorem ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$; therefore, ${{\sin }^{2}}\theta =1-{{\cos }^{2}}\theta $. Thus, we apply the theorem. And as per the quotient identity $\tan x=\frac{\sin x}{\cos x}$
$\begin{align}
& \frac{1-{{\cos }^{2}}x}{\cos x}=\frac{{{\sin }^{2}}x}{\cos x} \\
& =\frac{\sin x}{\cos x}.\sin x \\
& =\tan x.\sin x
\end{align}$
Therefore, the left side is equal to the right side.
