Answer
See the explanation below.
Work Step by Step
$-\ln \left| \cos x \right|$
Now, using the logarithm identity $\ln {{a}^{x}}=x\ln a$, the above expression can be simplified as,
$\begin{align}
& -\ln \left| \cos x \right|=\ln {{\left| \cos x \right|}^{-1}} \\
& =\ln \left| \frac{1}{\cos x} \right|
\end{align}$
By using the reciprocal identity, which is $\sec x=\frac{1}{\cos x}$ and simplifying further, we get:
$\ln \left| \frac{1}{\cos x} \right|=\ln \left| \sec x \right|$
Thus, the right-hand side of the expression is equal to the left-hand side, which is
$\ln \left| \sec x \right|=-\ln \left| \cos x \right|$.
Hence, it is proved that $\ln \left| \sec x \right|=-\ln \left| \cos x \right|$.
