Power Reduction Formulas/Hyperbolic Cosine Squared
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Theorem
- $\cosh^2 x = \dfrac {\cosh 2 x + 1} 2$
where $\cosh$ denotes hyperbolic cosine.
Proof 1
\(\ds 2 \cosh^2 x - 1\) | \(=\) | \(\ds \cosh 2 x\) | Double Angle Formula for Hyperbolic Cosine: Corollary $1$ | |||||||||||
\(\ds \cosh^2 x\) | \(=\) | \(\ds \frac {\cosh 2 x + 1} 2\) | solving for $\cosh^2 x$ |
$\blacksquare$
Proof 2
\(\ds \cosh^2 x\) | \(=\) | \(\ds \frac 1 4 \paren {e^x + e^{-x} }^2\) | 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 |
$\blacksquare$
Proof 3
\(\ds \cosh^2 x\) | \(=\) | \(\ds \cos^2 i x\) | Hyperbolic Cosine in terms of Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\map \cos {2 i x} + 1} 2\) | Square of Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\cosh 2 x + 1} 2\) | Hyperbolic Cosine in terms of Cosine |
$\blacksquare$
Also see
- Square of Hyperbolic Sine
- Cube of Hyperbolic Sine
- Cube of Hyperbolic Cosine
- Fourth Power of Hyperbolic Sine
- Fourth Power of Hyperbolic Cosine
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.37$: Powers of Hyperbolic Functions