Power Reduction Formulas/Hyperbolic Sine Squared
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Theorem
- $\sinh^2 x = \dfrac {\cosh 2 x - 1} 2$
where $\sinh$ and $\cosh$ denote hyperbolic sine and hyperbolic cosine respectively.
Proof 1
| \(\ds 2 \sinh^2 x + 1\) | \(=\) | \(\ds \cosh 2 x\) | Double Angle Formula for Hyperbolic Cosine: Corollary $2$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sinh^2 x\) | \(=\) | \(\ds \frac {\cosh 2 x - 1} 2\) | solving for $\sinh^2 x$ |
$\blacksquare$
Proof 2
| \(\ds \sinh^2 x\) | \(=\) | \(\ds \paren {\frac {e^x - e^{-x} } 2}^2\) | Definition of Hyperbolic Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 4 \paren {e^{2 x} + e^{-2 x} - 2}\) | multiplying out | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\dfrac {e^{2 x} + e^{-2 x} } 2 - 1}\) | rearranging | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\cosh 2 x - 1} 2\) | Definition of Hyperbolic Cosine |
$\blacksquare$
Also see
- Square of Hyperbolic Cosine
- 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: $8.36$: Powers of Hyperbolic Functions