Answer
$a.\quad $see below
$ b.\quad$ see below
Work Step by Step
$ a.\quad$
On the left side of the graph, the line segment drawn with a full line
has length$\quad \pi-\sec^{-1}(-x)$
On the right side, the line segment has length $\sec^{-1}x$
Symmetry suggests that these lengths are equal, that is
$\pi-\sec^{-1}(-x)=\sec^{-1}x\qquad$ ... or adding $\sec^{-1}(-x)-\sec^{-1}x,$
$\sec^{-1}(-x)=\pi-\sec^{-1}x$
$ b.\quad$
$\displaystyle \sec^{-1}(-x)=\cos^{-1}(\frac{1}{-x})\qquad$ ... by eq.5
$=\displaystyle \cos^{-1}(-\frac{1}{x}) = \pi -\cos^{-1}(\frac{1}{x}) \qquad$ ... by eq.3
$=\pi -\sec^{-1}(x) \qquad$ ... by eq.5
