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JEE Advanced 2023Paper 1
Exam
JEE Advanced
Year
2023
Paper
Paper 1
Subject
Mathematics
Chapter
Trigonometric Equations

JEE Advanced 2023 · Paper 1 · Mathematics · Trigonometric Equations

Let ^ - 1 (x) ∈ (- (π)/(2), (π)/(2)) , for x ∈ ℝ . Then the number...

Let tan⁡−1(x)∈(−π2,π2)\tan^{- 1}\left(x\right) \in \left(- \dfrac{\pi}{2}, \dfrac{\pi}{2}\right), for x∈Rx \in \mathbb{R}. Then the number of real solutions of the equation 1+cos⁡(2x)=2tan⁡−1(tan⁡x)\sqrt{1 + \cos \left(2x\right)} = \sqrt{2}\tan^{- 1}\left(\tan x\right) in the set \left(- \dfrac{3\pi}{2}, - \dfrac{\pi}{2}\right) \cup (−π2,π2)∪(π2,3π2)\left(- \dfrac{\pi}{2}, \dfrac{\pi}{2}\right) \cup \left(\dfrac{\pi}{2}, \dfrac{3\pi}{2}\right) is equal toLet ^ - 1 (x) ∈ (- (π)/(2), (π)/(2)) , for x ∈ ℝ . Then the number of real solutions of the equation 1 + (2x) = 2 ^ - 1

Correct answer

0

Step-by-step solution

Given,

1+cos⁡2x=2tan⁡−1(tan⁡x)</p><p>\sqrt{1 + \cos 2x} = \sqrt{2}\tan^{- 1}\left(\tan x\right)</p><p>⇒ \sqrt{2\cos^{2}x} = \sqrt{2}\tan^{- 1}\left(\tan x\right)</p><p></p><p>⇒ \left|\cos x\right| = tan⁡−1(tan⁡x)\tan^{- 1}\left(\tan x\right)

Now plotting the diagram of \left|\cos x\right| \text{ and } \tan^{- 1}\left(\tan x\right) we get,

Let ^ - 1 (x) ∈ (- (π)/(2), (π)/(2)) , for x ∈ ℝ . Then the number of real solutions of the equation 1 + (2x) = 2 ^ - 1

Now from above diagram we can say that,

Number of solutions = Number of intersection points = 3

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