Question #41659

Let $f(x) = \sin^{-1}(\sin x) + \cos^{-1}(\cos x)$. Let $C$ be the curve $y = f(x)$ in the $xy$-plane. Let $\vec{v}_1$ be the limiting unit tangent vector to $C$ as $x \to 0^+$, such that $\vec{v}_1 \cdot \hat{i} > 0$. Let $\vec{v}_2$ be the limiting unit normal vector to $C$ as $x \to 0^-$, such that $\vec{v}_2 \cdot \hat{j} > 0$. Let $P_1$ be the plane passing through the origin with normal vector $\vec{n}_1 = \vec{v}_1 \times \hat{k}$, and $P_2$ be the plane passing through the origin with normal vector $\vec{n}_2 = \vec{v}_2 \times (\hat{i} + \hat{k})$. If $\theta$ is the acute angle between the planes $P_1$ and $P_2$, then the value of $\cos \theta$ is: