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Let \( \tan^{-1}(x) \in \left(-\frac{\pi}{2},\frac{\pi}{2}\right) \), for \( x \in \mathbb{R} \). Then the number of real solutions of the equations \(\sqrt{1+\cos(2x)}=\sqrt{2}\,\tan^{-1}(\tan x) \) in the set \( \left(-\frac{3\pi}{2},\frac{\pi}{2}\right)\cup\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\cup\left(\frac{\pi}{2},\frac{3\pi}{2}\right) \) is equal to _____.

Considering only the principal values of the inverse trigonometric functions, the value of
\( \dfrac{3}{2} \cos^{-1}\sqrt{\dfrac{2}{2+\pi^{2}}} + \dfrac{1}{4} \sin^{-1}\dfrac{2\sqrt{2}\,\pi}{2+\pi^{2}} + \tan^{-1}\dfrac{\sqrt{2}}{\pi} \) is ____________.

For any \( y \in \mathbb{R} \), let \( \cot^{-1}(y) \in (0, \pi) \) and \( \tan^{-1}(y) \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \). Then the sum of all the solutions of the equation \( \tan^{-1}\!\left( \frac{6y}{9-y^{2}} \right) + \cot^{-1}\!\left( \frac{9-y^{2}}{6y} \right) = \frac{2\pi}{3} \) for \( 0 < |y| < 3 \), is equal to:

Considering only the principal values of the inverse trigonometric functions, the value of \( \tan\!\left(\sin^{-1}\!\left(\frac{3}{5}\right)-2\cos^{-1}\!\left(\frac{2}{\sqrt{5}}\right)\right) \) is

The total number of real solutions of the equation \( \theta = \tan^{-1}(2 \tan \theta) - \dfrac{1}{2}\sin^{-1}\!\left(\dfrac{6 \tan \theta}{9 + \tan^{2}\theta}\right) \) is
(Here, the inverse trigonometric functions \( \sin^{-1} x \) and \( \tan^{-1} x \) assume values in \( \left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right] \) and \( \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right) \), respectively.)

For any positive integer \( n \), let \( S_{n} : (0,\infty) \to \mathbb{R} \) be defined by
\( S_{n}(x) = \displaystyle \sum_{k=1}^{n} \cot^{-1}\!\left(\dfrac{1 + k(k+1)x^{2}}{x}\right) \)
Where for any \( x \in \mathbb{R} \), \( \cot^{-1}(x) \in (0,\pi) \) and \( \tan^{-1}(x) \in \left( -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right) \). Then which of the following statements is (are) TRUE?

Considering only the principal values of the inverse trigonometric functions, the value of
\( \cot^{-1}(\cot(-11)) + 10\sin\!\left(2\cos^{-1}\!\left(\frac{1}{\sqrt{2}}\right)\right) + 10\sin\!\left(2\tan^{-1}(2)\right) \) is:

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JEE Advanced 2026 Paper - 1

Considering only the principal values of the inverse trigonometric functions, the value of
\( \cot^{-1}(\cot(-11)) + 10\sin\!\left(2\cos^{-1}\!\left(\frac{1}{\sqrt{2}}\right)\right) + 10\sin\!\left(2\tan^{-1}(2)\right) \) is: