Power Reduction Formulas/Hyperbolic Cosine to 4th/Proof 2
Jump to navigation
Jump to search
Theorem
- $\cosh^4 x = \dfrac {3 + 4 \cosh 2 x + \cosh 4 x} 8$
Proof
\(\ds \cosh^4 x\) | \(=\) | \(\ds \frac 1 {2^4}\left(e^{x} + e^{-x}\right)^4\) | Definition of Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {16} \left({e^{4 x} + 4 e^{2 x} + 6 e^{0 x} + 4 e^{-2 x} + e^{-4 x} }\right)\) | Binomial Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 8 \left({\frac{e^{4 x} + e^{-4 x} } 2}\right) + \frac 4 8 \left({\frac{e^{2 x} + e^{-2 x} } 2 }\right) + \frac 6 {16}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 + 4 \cosh 2 x + \cosh 4 x} 8\) | Definition of Hyperbolic Cosine |
$\blacksquare$