Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $2 \sin ^{-1} x + 3 \cos ^{-1} x = \frac{2 \pi}{5}$ is:

If $\sin ^{-1} a=\alpha+\beta$ and $\sin ^{-1} b=\alpha-\beta$ then,$\sin ^2 \alpha+\cos ^2 \beta=$ . . . . . . .

If $\tan^{-1} 2x + \tan^{-1} 3x = \frac{\pi}{4}$,then $x =$

If $\tan ^{-1} 2 x+\tan ^{-1} 3 x=\frac{\pi}{4}$,then the value of $x$ is

The derivative of $\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ with respect to $x$ is:

If $S = \{x \in R : \sin^{-1}\left(\frac{x+1}{\sqrt{x^2+2x+2}}\right) - \sin^{-1}\left(\frac{x}{\sqrt{x^2+1}}\right) = \frac{\pi}{4}\}$,then $\sum_{x \in S} \left(\sin\left((x^2+x+5)\frac{\pi}{2}\right) - \cos((x^2+x+5)\pi)\right)$ is equal to $........$.