Question #41754

Let $P_1$ be a plane with normal vector $\vec{n}_1 = a\hat{i} + b\hat{j} + c\hat{k}$, where $a, b,$ and $c$ are the coefficients of $x^2, x^3,$ and $x^4$ respectively in the expansion of $(1-x)^{-3}$ for $|x| < 1$. Let $P_2$ be another plane with a unit normal vector $\vec{n}_2 = p\hat{i} + q\hat{j} + r\hat{k}$, where $p, q, r \in \mathbb{R}$ satisfy the equation $\sin^{-1}(p) + \cos^{-1}(q) + \csc^{-1}(r) = \pi$. If $\theta$ is the acute angle between the planes $P_1$ and $P_2$, then the value of $19 \cos \theta$ is equal to