The following occurred while working on some research project. Since the methods of proof I used were lengthy, I wish to see a skillful or insightful approach (perhaps even conceptual). Anyhow, here it is. Let 
$$f(x)=\left(\frac{x}{e^x-1}\right)^2 + \left(\frac{x+1}{e^{x+1}+1}\right)^2.$$
Can one give a short and elegant proof of these statements?

(1) $f(x)$ is a strictly decreasing function of $x$ over $\mathbb{R}$.

(2) In fact, the statement holds true if $e$ is replaced by any real number $t>1$.