Hyperbolic Sine Function is Odd

Theorem
Let $x \in \R$ be a real number.

Let $\sinh x$ be the hyperbolic sine of $x$.

Then:
 * $\sinh \left({-x}\right) = -\sinh x$

That is, the hyperbolic sine function is odd.

Proof
Recall the definition of the hyperbolic sine function:


 * $ \displaystyle \sinh x = \frac {e^{x} - e^{-x}} 2 $

Then,