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 |
$\blacksquare$