An Even function divided by Odd function is an Odd function,that is a fact. However is there a means to prove this?
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$\begingroup$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).$$\text{ }$ $\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\square$
$\endgroup$ $\begingroup$Let $f$ be an odd function : $ f(-x) = - f(x) $ for all $x$. Let $g$ be an even function: $g(-x) = g(x) $ for all $x$. Now, put $h(x) = \frac{ g(x) }{f(x) } $
$$ h(-x) = \frac{ g(-x) }{f(-x) } = \frac{ g(x) }{ - f(x) } = - \frac{ g(x) }{f(x) }= - h(x) $$
Hence, $h$ is odd.
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