Find $\varpi$ s.t. $x\varpi\cot(\varpi(x-1))=0$ Iff $x$ is even.

Question

I was playing around with functions to find one that looks like $H_xH_{-x}$ (where H is the analytic continuation of the harmonic numbers) to find H’s zeroes. That is when I found that (based on numerical evidence) the second zero of H satisfies: $$x\varpi\cot(\varpi(x-1))=0$$ if $x$ is an even integer. I tried solving this by dividing the coefficient of the cotangent function on both sides and then taking inverse cotangent and then solving. But when I do that:$$\cot(\varpi(x-1))=0$$$$\varpi(x-1)=\cot(0)$$$$\varpi=\frac{\pi}{2(x-1)}$$, I get a set of values and setting $x=0.5$ which gives the correct answer of $\pi/2$ even though $x$ can’t be non-integer. Is this a correct way to solving this problem?