Answer
See work below.
Work Step by Step
Odd functions: $f(-x) = -f(x)$.
$f(x)=\frac{1-e^{\frac{1}{x}}}{1+e^{\frac{1}{x}}}$
$f(-x)=\frac{1-e^{\frac{1}{-x}}}{1+e^{\frac{1}{-x}}}$
$-f(x)=\frac{-1+e^{\frac{1}{x}}}{1+e^{\frac{1}{x}}}$
Let's take $f(-x)$
$f(-x)=\frac{1-e^{\frac{1}{-x}}}{1+e^{\frac{1}{-x}}}$
$=\frac{1-\frac{1}{e^{\frac{1}{x}}}}{1+\frac{1}{e^{\frac{1}{x}}}}$
$=\left(\frac{1-\frac{1}{e^{\frac{1}{x}}}}{1+\frac{1}{e^{\frac{1}{x}}}}\right)\left(\frac{e^\frac{1}{x}}{e^{\frac{1}{x}}}\right)$
$=\frac{e^{\frac{1}{x}}-1}{e^{\frac{1}{x}}+1}$
$=-f(x)$
