Power Reduction Formulas/Hyperbolic Cosine Squared/Proof 2

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Theorem

$\cosh^2 x = \dfrac {\cosh 2 x + 1} 2$


Proof

\(\ds \cosh^2 x\) \(=\) \(\ds \frac 1 4 \paren {e^x + e^{-x} }^2\) Definition of Hyperbolic Cosine
\(\ds \) \(=\) \(\ds \frac {e^{2 x} + e^{-2 x} + 2} 4\)
\(\ds \) \(=\) \(\ds \frac {\cosh 2 x + 1} 2\) Definition of Hyperbolic Cosine

$\blacksquare$