Let S= θ ∈[0,2 π): tan (π cos θ)+ tan (π sin θ)=0 . Then displaystyle∑θ ∈ s sin 2(θ+(π/4)) is equal to

Question

Let S= θ ∈[0,2 π): tan (π cos θ)+ tan (π sin θ)=0 . Then displaystyle∑θ ∈ s sin 2(θ+(π/4)) is equal to  (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

tan (π cos θ)+ tan (π sin θ)=0 tan (π cos θ)=- tan (π sin θ) tan (π cos θ)= tan (-π sin θ) π cos θ= n π-π sin θ sin θ+ cos θ= n text where n ∈ I possible values are n =0,1 and -1 because -√2 ≤ sin θ+ cos θ ≤ √2 Now it gives θ ∈ 0, (π/2), (3 π/4), (7 π/4), (3 π/2), π So displaystyle∑θ ∈ S sin 2(θ+(π/4))=2(0)+4((1/2))=2

Q. Let S={θ∈[0,2π):tan(πcosθ)+tan(πsinθ)=0}. Then θ∈s∑​sin2(θ+4π​) is equal to _______

Q. Let $S = {θ \in [0, 2 π) : tan (π cos θ) + tan (π sin θ) = 0}$ .
Then $θ \in s \sum sin^{2} (θ + \frac{π}{4})$ is equal to _______