Let S = { [ 0 , 2 ) : tan ( cos ) + tan ( sin ) = 0 } . Then S sin 2 ( + 4 ) is equal to ________ . [2023]

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Solution

Q.
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 ________ . [2023]

Ans.
(2)

$Given, S = {θ \in [0, 2 π) : \tan (π \cos θ) + \tan (π \sin θ) = 0}$

$Now, \tan (π \cos θ) + \tan (π \sin θ) = 0$

$\Rightarrow \tan (π \cos θ) = - \tan (π \sin θ)$

$\Rightarrow \tan (π \cos θ) = \tan (- π \sin θ)$

$∴$ $π \cos θ = n π - π \sin θ$

$∴$ $\sin θ + \cos θ = n,$ where, $n \in I$

Possible values are $n = 0, 1, - 1$ because

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

$Now, it gives θ \in {0, \frac{π}{2}, \frac{3 π}{4}, \frac{7 π}{4}, \frac{3 π}{2}, π}$

$So, \sum_{θ \in S} \sin^{2} (θ + \frac{π}{4}) = 2 (0) + 4 (\frac{1}{2}) = 2$

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