3d Geometry

Let \(\theta\) be the angle between the planes \(P_{1}: \vec{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})=9\) and \(P_{2}: \vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=15\). Let \(\mathrm{L}\) be the line that meets \(P_{2}\) at the point \((4,-2,5)\) and makes an angle \(\theta\) with the normal of \(P_{2}\). If \(\alpha\) is the angle between \(\mathrm{L}\) and \(P_{2}\), then \(\left(\tan ^{2} \theta\right)\left(\cot ^{2} \alpha\right)\) is equal to ____________.