Evaluate $\sum_{n=1}^\infty \arctan\frac{1}{n^2}$

Question

As shown here (Does $\sum_{n=1}^{\infty}\arctan(\frac{1}{n^2})$ converge?), the series $\sum_{n=1}^\infty \arctan\frac{1}{n^2}$ converges. It is similar to $$\sum_{n=1}^\infty \arctan\frac{2}{n^2}=\frac{3\pi}{4}$$ (Evaluate $\sum_{n=1}^{\infty}\arctan(\frac{2}{n^2})$?). A computer algebra system suggests that $$\sum_{n=1}^\infty \arctan\frac{1}{n^2}=\arctan \frac{1-\cot\frac{\pi}{\sqrt{2}}\tanh\frac{\pi}{\sqrt{2}}}{1+\cot\frac{\pi}{\sqrt{2}}\tanh\frac{\pi}{\sqrt{2}}}.$$ If it's true, how can I prove it?