Question

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 _______

Answer 1

The correct answer is 2
tan(πcosθ)+tan(πsinθ)=0
tan(πcosθ)=−tan(πsinθ)
tan(πcosθ)=tan(−πsinθ)
πcosθ=nπ−πsinθ
sinθ+cosθ=n where n∈I
possible values are n=0,1 and −1 because
−2​≤sinθ+cosθ≤2​
Now it gives θ∈{0,2π​,43π​,47π​,23π​,π}
So θ∈S∑​sin2(θ+4π​)=2(0)+4(21​)=2