Let θ be the angle between the planes P_1: r ·( i + j +2 k )=9 and P_2: r ·(2 i - j + k )=15. Let L be the line that meets P_2 at the point (4,-2,5)…

Let $\displaystyle \theta$ be the angle between the planes $\displaystyle P_1: \vec{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})=9$ and $\displaystyle P_2: \vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=15$. Let L be the line that meets $\displaystyle P_2$ at the point $\displaystyle (4,-2,5)$ and makes an angle $\displaystyle \theta$ with the normal of $\displaystyle P_2$. If $\displaystyle \alpha$ is the angle between L and $\displaystyle P_2$, then $\displaystyle \left(\tan ^2 \theta\right)\left(\cot ^2 \alpha\right)$ is equal to $\displaystyle \_\_\_\_$.