$\cot ^{-1}\left(2 \cdot 1^{2}\right)+\cot ^{-1}\left(2 \cdot 2^{2}\right)+\cot ^{-1}\left(2 \cdot 3^{2}\right)+\ldots$ up to $\infty$ is equal to

If $k = \tan(\frac{\pi}{4} + \frac{1}{2}\cos^{-1}(\frac{2}{3})) + \tan(\frac{1}{2}\sin^{-1}(\frac{2}{3}))$, then the number of solutions of the equation $\sin^{-1}(kx-1) = \sin^{-1}x - \cos^{-1}x$ is . . . . . . .

If ${\sin ^{ - 1}}a + {\sin ^{ - 1}}b + {\sin ^{ - 1}}c = \pi ,$ then the value of $a\sqrt {1 - {a^2}} + b\sqrt {1 - {b^2}} + c\sqrt {1 - {c^2}}$ is

If $x \in [0, 1]$,then the number of solution$(s)$ of the equation $2[\cos^{-1}x] + 6[\text{sgn}(\sin x)] = 3$ is (where $[.]$ denotes the greatest integer function and $\text{sgn}(x)$ denotes the signum function of $x$)-

$\sin ^{-1}\left(\frac{12}{13}\right)+\cos ^{-1}\left(\frac{4}{5}\right)+\tan ^{-1}\left(\frac{63}{16}\right)=$

The number of positive integral solutions of $\tan ^{-1} x+\cos ^{-1}\left(\frac{y}{\sqrt{1+y^2}}\right)=\sin ^{-1}\left(\frac{3}{\sqrt{10}}\right)$ are