1. ## Hyperbolic Functions

1.) Show that $\displaystyle \cosh ^2 x - \sinh ^2 x = 1$ for all $\displaystyle x$

2.) Find all solutions of $\displaystyle \sinh (x^2 - 1) = 0$

Thanks again guys.

2. Originally Posted by toop
1.) Show that $\displaystyle \cosh ^2 x - \sinh ^2 x = 1$ for all $\displaystyle x$
use the fact that $\displaystyle \sinh x = \frac {e^x - e^{-x}}2$ and $\displaystyle \cosh x = \frac {e^x + e^{-x}}2$

2.) Find all solutions of $\displaystyle \sinh (x^2 - 1) = 0$
just set $\displaystyle x^2 - 1 = 0$

to see why this is a solution (that is, why the solution to $\displaystyle \sinh x = 0$ is $\displaystyle x = 0$) you can use the definition above

3. Can you expand a little further on #1...I have no idea how to do it.

And for #2, I came out with $\displaystyle x=1$ ....does that look correct?

4. Originally Posted by toop
Can you expand a little further on #1...I have no idea how to do it.
$\displaystyle \cosh^2 x - \sinh^2 x = \left( \frac {e^x + e^{-x}}2 \right)^2 - \left( \frac {e^x - e^{-x}}2 \right)^2 = ...$

if you are familiar with complex numbers, there is another approach...but this is pretty straight forward, so this is good enough. plus, it gives you more stuff to write so you can show of your elegance in writing math

And for #2, I came out with $\displaystyle x=1$ ....does that look correct?
that is only one solution, there are two

5. Oops, missed that... $\displaystyle x = -1, 1$

And I think I got #1...thanks!