On polygamma reflection formula

In this thread we are proving the polygamma reflection formula stating that:

$$\psi^{(n)}(1-z)+(-1)^{n+1}\psi^{(n)}(z)=(-1)^n  \pi \frac{\mathrm{d}^n }{\mathrm{d} x^n}\cot \pi z$$

 
Proof:

It is sufficient to prove that

$$\begin{equation} \psi(1-z)-\psi(z)=\pi \cot \pi z  \end{equation}$$

since then we can differentiate $(1)$ multiple times and get the formula above.

Let's kick things off by starting with the Weierstraß product of the $\Gamma$ function, stating that:
$$\Gamma(t+1) = e^{-\gamma t}\prod_{n=1}^{\infty}\left(1+\frac{t}{n}\right)^{-1}e^{\frac{t}{n}}$$

leading us to the reflection formula of $\Gamma$, that is the formula $\displaystyle  \Gamma (1-z)\Gamma (z)=\pi \csc \pi z$. Taking $\ln $ at the reflection formula and differentiating we get:

$$\psi(z)-\psi(1-z) = \frac{d}{dz}\,\log\sin(\pi z) \Rightarrow \psi(1-z)-\psi(z)=\pi \cot \pi z$$

and $(1)$ is proved.

Now, differentiating multiple times $(1)$ we get the polygamma reflection formula. We prove it using induction.