# Even function divided by Odd function is an Odd function PROOF?

An Even function divided by Odd function is an Odd function,that is a fact. However is there a means to prove this?

Let $h(x)$ be an even function, which means that $h(-x)=h(x)$ and $g(x)$ be an odd function, which means that $g(-x)=-g(x)$.
Let $f(x)=\frac{h(x)}{g(x)}$.
Then, $f(-x)=\frac{h(-x)}{g(-x)}=\frac{h(x)}{-g(x)}=-f(x).$$\text{ } \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\square Let f be an odd function : f(-x) = - f(x) for all x. Let g be an even function: g(-x) = g(x) for all x. Now, put h(x) = \frac{ g(x) }{f(x) }$$ h(-x) = \frac{ g(-x) }{f(-x) } = \frac{ g(x) }{ - f(x) } = - \frac{ g(x) }{f(x) }= - h(x)$$Hence,$h\$ is odd.