Answer
The function is odd. Notice that the function is symmetric with respect to the origin.
$f(-x)=-f(x)$, that is $-x+tan(-x)=-(x+tan~x)$
Work Step by Step
$f(x)=x+tan~x$
$f(-x)=-x+tan(-x)=-x+\frac{sin(-x)}{cos(-x)}=-x+\frac{-sin~x}{cos~x}=-x-tan~x=-(x+tan~x)=-f(x)$
