Simple Question About Expectation of a Product with Quotients

Question

Let $X(k)$ be i.i.d. random variables with arbitrary density $f_X$ and mean $\mathbb{E}[X(k)] =\mu$ for $k=0,1,...,N-1$. Let $a \in \mathbb{R}^1$, I would like to compute $$ \mathbb{E}\left[ {\prod\limits_{k = 0}^{N - 1} {\frac{{1 - aX\left( k \right)}}{{1 + aX\left( k \right)}}} } \right]. $$ Below is my thinking: \begin{align*} \mathbb{E}\left[ {\prod\limits_{k = 0}^{N - 1} {\frac{{1 - aX\left( k \right)}}{{1 + aX\left( k \right)}}} } \right] &= \prod\limits_{k = 0}^{N - 1} {\mathbb{E}\left[ {\frac{{1 - aX\left( k \right)}}{{1 + aX\left( k \right)}}} \right]} \hfill \\ &?= \prod\limits_{k = 0}^{N - 1} {\frac{{\mathbb{E}\left[ {1 - aX\left( k \right)} \right]}}{{\mathbb{E}\left[ {1 + aX\left( k \right)} \right]}}} \hfill \\ &= \prod\limits_{k = 0}^{N - 1} {\frac{{1 - a\mathbb{E}\left[ {X\left( k \right)} \right]}}{{1 + a\mathbb{E}\left[ {X\left( k \right)} \right]}}} \hfill \\ &= {\left( {\frac{{1 - a\mu }}{{1 + a\mu }}} \right)^N} \hfill \\ \end{align*} But I am somewhat bothers with my second equality above since I am not clear if it is allowed to take expectation on the denominator and numerator separately under i.i.d. condition. (I was thinking perhaps using iterated conditional expectation + i.i.d. might help me out). Any suggestion is appreciated.