A few procrastinal computations motivated by Four infinite series involving Riemann zeta function suggest the identity $$\tan\left(\frac{\kappa-1}{\kappa+1}\frac{\pi}{2}\right)=\frac{1}{\pi}\sum_{n=1}^\infty \frac{\kappa^n-1}{(\kappa+1)^n}\zeta(n+1)$$ for all $\kappa>0$. (Both sides change sign when replacing $\kappa$ by $1/\kappa$. It is thus enough to consider $\kappa\geq 1$.)

I checked it on $100$ random values up to 400 digits which convinced me that it must hold.

Is there an easy explanation?

(Remark: This identity, if true, provides of course easy answers to question M0401468 mentionned above.)

(Computational remark: For high precision computation, it is perhaps best to cut the sum at some not very large value $N$ (I worked with $N=400$) and to approximate the omitted terms by summing the first few terms contributing to $\zeta(N+1),\ldots$, by considering the associated geometric progressions.)