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