When is $\sum_{i=1}^{\infty}\frac{(-1)^{i+1}x^{2i-1}}{1-x^{2i-1}}=\sum_{i=1}^{\infty}\frac{x^{i}}{1+x^{2i}}$ useful?

Question

I recently came across this rather curious identity for $|x| < 1$: $$\sum_{i=1}^{\infty}\frac{(-1)^{i+1}x^{2i-1}}{1-x^{2i-1}}=\sum_{i=1}^{\infty}\frac{x^{i}}{1+x^{2i}}$$ The proof is fairly straightforward, so I won't go into details about it. I was wondering if anyone could give some examples of exercises where this identity is used (e.g. any other series that can be evaluated or simplified using this identity), as I feel it would be interesting to explore some questions relating to this.