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Let \( x = \sin\!\left(2 \tan^{-1}\!\alpha\right) \) and \( y = \sin\!\left(\dfrac{1}{2}\tan^{-1}\!\dfrac{4}{3}\right) \). If \( S = \{ \alpha \in \mathbb{R} : y^2 = 1 - x \} \); then \( \displaystyle\sum_{\alpha \in S} 16\alpha^3 \) is equal to __________.

For \( k \in \mathbb{R} \), let the solutions of the equation \( \cos\!\left(\sin^{-1}\!\left(\operatorname{x cot}\!\left(\tan^{-1}\!\left(\cos\!\left(\sin^{-1} x\right)\right)\right)\right)\right) = k, \; 0<|x|<\dfrac{1}{\sqrt{2}} \) be \( \alpha \) and \( \beta \), where the inverse trigonometric functions take only principle values. If the solution of the equation \( x^2 - bx - 5 = 0 \) are \( \dfrac{1}{\alpha^2} + \dfrac{1}{\beta^2} \) and \( \dfrac{\alpha}{\beta} \) then \( \dfrac{b}{k^2} \) is equal to _______

\( \tan \left( 2\tan^{-1}\dfrac{1}{5}+\sec^{-1}\dfrac{\sqrt{5}}{2}+2\tan^{-1}\dfrac{1}{8} \right) \) is equal to:

The set of all values of \( k \) for which \( (\tan^{-1}x)^{3} + (\cot^{-1}x)^{3} = k\pi^{3},\ x \in \mathbb{R} \), is the interval:

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation \( \cos^{-1}(x)-2\sin^{-1}(x)=\cos^{-1}(2x) \) is equal to:

Let \( x^* y = x^2 + y^3 \) and \( (x^ * 1) ^* 1 = x^* (1 ^* 1) \). Then a value of \( 2 \sin^{-1}\!\left(\frac{x^4 + x^2 - 2}{x^4 + x^2 + 2}\right) \) is:

If \( 0 < x < \dfrac{1}{\sqrt{2}} \text{ and } \dfrac{\sin^{-1} x}{\alpha} = \dfrac{\cos^{-1} x}{\beta}\) then the value of \(\sin\left(\dfrac{2\pi\alpha}{\alpha+\beta}\right)\) is:

Considering only the principal values of the inverse trigonometric functions, the domain of the function \( f(x)=\cos^{-1}\!\left(\dfrac{x^2-4x+2}{x^2+3}\right) \) is:

\( \sin^{-1}\!\left(\sin\frac{2\pi}{3}\right)+\cos^{-1}\!\left(\cos\frac{7\pi}{6}\right)+\tan^{-1}\!\left(\tan\frac{3\pi}{4}\right) \text{ is equal to:} \)

The value of \( \cot\!\left( \displaystyle \sum_{n=1}^{50} \tan^{-1}\!\left(\frac{1}{1+n+n^{2}}\right)\right) \) is

The value of \( \tan^{-1}\!\left( \frac{\cos\!\left( \frac{15\pi}{4} \right)}{\sin\!\left( \frac{\pi}{4} \right)} - 1 \right) \) is equal to:

\( 50 \tan\!\left(3\tan^{-1}\!\left(\dfrac{1}{2}\right) + 2\cos^{-1}\!\left(\dfrac{1}{\sqrt{5}}\right)\right) + 4\sqrt{2}\,\tan\!\left(\dfrac{1}{2}\tan^{-1}(2\sqrt{2})\right) \) is equal to __________.

If the inverse trigonometric functions take principal values, then \( \cos^{-1}\!\left(\frac{3}{10}\cos\!\left(\tan^{-1}\!\left(\frac{4}{3}\right)\right)+\frac{2}{5}\sin\!\left(\tan^{-1}\!\left(\frac{4}{3}\right)\right)\right) \) is equal to:

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JEE Main 2022 26th July Evening Shift

If \( 0 < x < \dfrac{1}{\sqrt{2}} \text{ and } \dfrac{\sin^{-1} x}{\alpha} = \dfrac{\cos^{-1} x}{\beta}\) then the value of \(\sin\left(\dfrac{2\pi\alpha}{\alpha+\beta}\right)\) is: