I heuristically discovered the following identity for the trigamma function, that I could not find in any tables or papers or infer from existing formulae (e.g. [1], [2], [3], [4], [5], [6]):
$$4\,\psi_1\!\left(\frac15\right)+\psi_1\!\left(\frac25\right)-\psi_1\!\left(\frac1{10}\right)=\frac{4\pi^2}{\phi\,\sqrt5}.\tag1$$
It also seems to be unknown to *Mathematica*, but numerically checks with at least $20000$ decimal digits. It might be provable through some application of reflection and multiplication theorems, but I couldn't do this.

Please suggest how to prove it.

*Update:*
Another identity is
$$3\,\psi_1\!\left(\frac1{12}\right)-30\,\psi_1\!\left(\frac13\right)=120\,G+\left(6\sqrt3-8\right)\pi^2.\tag2$$