(a) Starting from the definitions of tanh and sech in terms of exponentials, prove that
\[1-\tanh ^{2} x=\operatorname{sech}^{2} x\]
(b) Using the substitution \(u=\tanh x\), or otherwise, find \(\int \operatorname{sech}^{2} x \tanh ^{2} x \mathrm{~d} x\).
It is given that, for \(n \geqslant 0, I_{n}=\int_{0}^{\ln 3} \operatorname{sech}^{n} x \tanh ^{2} x \mathrm{~d} x\).
(c) Show that, for \(n \geqslant 2\),
\[(n+1) I_{n}=\left(\frac{4}{5}\right)^{3}\left(\frac{3}{5}\right)^{n-2}+(n-2) I_{n-2}\]
[You may use the result that \(\frac{\mathrm{d}}{\mathrm{d} x}(\operatorname{sech} x)=-\tanh x \operatorname{sech} x\).]
(d) Find the value of \(I_{4}\).