(a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that
\(2 \cosh ^{2} x=\cosh 2 x+1\)
(b) Find the solution of the differential equation
\(\frac{\mathrm{d} y}{\mathrm{~d} x}+2 y \tanh x=1\)
for which \(y=1\) when \(x=0\). Give your answer in the form \(y=\mathrm{f}(x)\).