Let $S = \{\theta \in [0, 2\pi): \tan(\pi\cos\theta) + \tan(\pi\sin\theta) = 0\}$. Then $\displaystyle\sum_{\theta \in S} \sin^2\left(\theta + \frac{\pi}{4}\right)$ is equal to
Step-by-Step Solution
Key Concept: The equation reduces to $\sin\theta + \cos\theta = n$ for integer $n$. Since $-\sqrt{2} \leq \sin\theta+\cos\theta \leq \sqrt{2}$, only $n=0,1,-1$ are possible. List all $\theta$ in $[0,2\pi)$ and compute sum.
$S = \{0, \pi/2, 3\pi/4, 7\pi/4, 3\pi/2, \pi\}$: $\sin^2(\pi/4)=1/2$, $\sin^2(3\pi/4)=1/2$, etc. $\sum = 2(0)+4(1/2) = 2$. Answer: 2
Correct Answer: 2