Nice proof! You could take any convergent infinite sum $\sum \frac{n_i}{d_i}$ where  $\gcd(n_i,d_i)=1$ , and for every integer $q \gt 1$ there is a $j$ so that $q$ divides $d_i$ for all $i \gt j$. For example $d_i$ is the least common multiple of the integers up to $i$ and $n_i$ is the first prime greater than $i$.The question is if there are nice reals with such a series.

I suppose that $\cos(1)=1-1/2+1/24-1/720\cdots$ works although $\cos(1)=\frac{e^i+e^{-i}}{2}$ so it depends what counts as a trivial relation. Also, $\sec(1)$ looks ok.