QJEEM23Ma60e5: JEE Main 2023 Mathematics - Class 11 JEEM23Ma60e5 Solution: Let S = { θ ∈ [ 0 , 2 π ) : tan ⁡ ( π cos ⁡ θ ) + tan ⁡ ( π sin …

Question

Verified Solution • 2025–26 Session moderate

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

Accepted Answer

\begin{aligned} \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 where n \in I \end{aligned}

possible values are

\mathrm{n}=0,1

and

-1

because

- \sqrt{2} \leq \sin θ + \cos θ \leq \sqrt{2}

Now it gives

\theta \in\left\{0, \frac{\pi}{2}, \frac{3 \pi}{4}, \frac{7 \pi}{4}, \frac{3 \pi}{2}, \pi\right\}

So

\sum \limits_{\theta \in \mathrm{S}} \sin ^2\left(\theta+\frac{\pi}{4}\right)=2(0)+4\left(\frac{1}{2}\right)=2

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