Problem of the Day #202

POTD June 29, 2024

Let $ABCD$ be the bottom face of a cube, and let $PQRS$ be the top face of that same cube, such that $AP, BQ, CR,$ and $DS$ form vertical edges. Let $\theta$ be the angle of intersection between space diagonals $AR$ and $BS$. If $\sin(\theta)+\cos(\theta)+\tan(\theta)$ can be expressed as $\frac{a+b\sqrt c}{d}$, where $a, b, c, d \in \mathbb{Z}^+$, $\gcd(a, b, d) = 1$, and $c$ is not divisible by the square of any prime, find $1000a+100b+10c+d$.