(a) Starting from the definitions of tanh and sech in terms of exponentials, prove that
\[1-\tanh ^{2} \theta=\operatorname{sech}^{2} \theta\]

The variables \(x\) and \(y\) are such that \(\tanh y=\cos \left(x+\frac{1}{4} \pi\right)\), for \(-\frac{1}{4} \pi<x<\frac{3}{4} \pi\).
(b) By differentiating the equation \(\tanh y=\cos \left(x+\frac{1}{4} \pi\right)\) with respect to \(x\), show that
\[\frac{\mathrm{d} y}{\mathrm{~d} x}=-\operatorname{cosec}\left(x+\frac{1}{4} \pi\right) .\]
(c) Hence find the first three terms in the Maclaurin's series for \(\tanh ^{-1}\left(\cos \left(x+\frac{1}{4} \pi\right)\right)\) in the form \(\frac{1}{2} \ln a+b x+c x^{2}\), giving the exact values of the constants \(a, b\) and \(c\).