For the equation $\cos^{-1} x + \cos^{-1} 2x + \pi = 0$,the number of real solutions is:

Let $f(x) = \cot \left( \sin^{-1} \sqrt{\frac{2}{3 + \cos 2x}} \right)$. Then,the value of $f'\left( \frac{2\pi}{3} \right)$ is:

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.

If $\alpha$ and $\beta$ are the roots of the equation $6x^2 - 5x + 1 = 0$,then the value of $\tan^{-1}\alpha + \tan^{-1}\beta$ is:

If $\tan ^{-1}(x+1)+\tan ^{-1} x+\tan ^{-1}(x-1)=\tan ^{-1} 3$,then for $x < 0$ the value of $500 x^4+270 x^2+997=$

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $16((\sec^{-1} x)^2 + (\operatorname{cosec}^{-1} x)^2)$ is: (in $\pi^2$)