Power Reduction Formulas/Hyperbolic Cosine to 4th/Proof 3
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Theorem
- $\cosh^4 x = \dfrac {3 + 4 \cosh 2 x + \cosh 4 x} 8$
Proof
| \(\ds \cosh^4 x\) | \(=\) | \(\ds \cos^4 i x\) | Hyperbolic Cosine in terms of Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {3 + 4 \cos \paren {2 i x} + \cos \paren {4 i x} } 8\) | Fourth Power of Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {3 + 4 \cosh 2 x + \cosh 4 x} 8\) | Hyperbolic Cosine in terms of Cosine |
$\blacksquare$