(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 \(y=\mathrm{f}(x)\).

(a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that
\(\sinh 2 x=2 \sinh x \cosh x .\)
(b) Using the substitution \(u=\sinh x\), find \(\int \sinh ^{2} 2 x \cosh x \mathrm{~d} x\).

(c) Find the particular solution of the differential equation
\(\frac{\mathrm{d} y}{\mathrm{~d} x}+y \tanh x=\sinh ^{2} 2 x,\)
given that \(y=4\) when \(x=0\). Give your answer in the form \(y=\mathrm{f}(x)\).

(a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that
\(\cosh 2 x=2 \sinh ^{2} x+1 .\)
(b) Find the set of values of \(k\) for which \(\cosh 2 x=k \sinh x\) has two distinct real roots.