If \( (\sin^{-1} x)^2 - (\cos^{-1} x)^2 = a; \, 0 < x < 1, \, a \neq 0 \), then the value of \( 2x^2 - 1 \) is:
\( \cos \left( \dfrac{2a}{\pi} \right) \)
\( \sin \left( \dfrac{4a}{\pi} \right) \)
\( \sin \left( \dfrac{2a}{\pi} \right) \)
\( \cos \left( \dfrac{4a}{\pi} \right) \)
If \( \displaystyle \sum_{r=1}^{50} \tan^{-1}\!\left(\frac{1}{2r^{2}}\right) = p \), then the value of \( \tan p \) is:
The domain of the function \( f(x) = \sin^{-1}\!\left(\dfrac{3x^{2}+x-1}{(x-1)^{2}}\right) + \cos^{-1}\!\left(\dfrac{x-1}{x+1}\right) \) is:
Let \( M \) and \( m \) respectively be the maximum and minimum values of the function \( f(x)=\tan^{-1}(\sin x+\cos x) \) in \( \left[0,\frac{\pi}{2}\right] \). Then the value of \( \tan(M-m) \) is equal to:
The sum of possible values of \( x \) for \( \tan^{-1}(x+1) + \cot^{-1}\!\left(\dfrac{1}{x-1}\right) = \tan^{-1}\!\left(\dfrac{8}{31}\right) \) is:
Given that the inverse trigonometric functions take principal values only. Then the number of real values of \( x \) which satisfy \( \sin^{-1}\!\left(\dfrac{3x}{5}\right)+\sin^{-1}\!\left(\dfrac{4x}{5}\right)=\sin^{-1}x \) is equal to:
The number of real roots of the equation \( \tan^{-1}\sqrt{x(x+1)} + \sin^{-1}\sqrt{x^{2}+x+1} = \dfrac{\pi}{4} \) is :
The value of \( \tan\!\left(2\tan^{-1}\!\left(\dfrac{3}{5}\right)+\sin^{-1}\!\left(\dfrac{5}{13}\right)\right) \) is equal to:
If the domain of the function \( f(x) = \dfrac{ \cos^{-1}\sqrt{x^2 - x + 1} }{ \sqrt{ \sin^{-1}\!\left( \dfrac{2x - 1}{2} \right) } } \) is the interval \( (\alpha, \beta) \), then \( \alpha + \beta \) is equal to:
If \( \cot^{-1}(\alpha) = \cot^{-1} 2 + \cot^{-1} 8 + \cot^{-1} 18 + \cot^{-1} 32 + \ldots \) upto \( 100 \) terms, then \( \alpha \) is:
