# Hyperbolic Function Identities

• Sep 12th 2009, 12:26 PM
Kasper
Hyperbolic Function Identities
Hey, I'm having a hell of a time trying to figure out this simplification; If anyone has any input, it would be greatly appreciated!

Quote:

Show that: $\frac{sinh(3t)}{sinh(t)} = 1 + 2 cosh(2t) \ for \ t \neq 0$
I've tried breaking the $\frac{sinh(3t)}{sinh(t)}$ side down by turning it into $\frac{sinh (2t+t)}{sinh t}$ and using the $sinh (t+u)$ identity, but everything i do just seems to complicate the whole deal by expanding it to huge proportions that doesn't chop down any.

Any ideas on a hot identity to try? I'm not sure how else to approach this.
• Sep 12th 2009, 01:14 PM
MPK
Hello,

try using : sh(3x) = 0.5 * (e^(3x) - e^(-3x))

sh(x) = 0.5 * (e^(x) - e^(-x))

sh(3x)/sh(x) = (e^(3x) - e^(-3x))/(e^(x) - e^(-x))

and then use the fact that :

a^3 -b^3 = (a-b)(a² + ab + b²)

and it will be paradise not hell (Wink) (Rofl)
• Sep 12th 2009, 02:20 PM
Kasper
Oh man I never thought to break it down into exponentials!

But I'm lost on how I can use $a^3 - b^3 = (a-b)(a^2 - ab + b^2)$ with $e^{3t} - e^{-3t}$ because of the negative exponent on the second $e$? Don't I have $e^{3t} - \frac{1}{e^{3t}}$?