If $\sin \left(\cot ^{-1}(x+1)\right)=\cos \left(\tan ^{-1} x\right)$,then considering positive square roots,$x$ has the value $\qquad$

$2 \tanh^{-1} \frac{1}{2}$ is equal to

The simplest form of $\cot ^{-1}\left(\frac{1}{\sqrt{x^2-1}}\right), x>1$ is . . . . . . .

Let $a, b$ and $c$ be the lengths of the sides of a triangle with its opposite angles $A, B$ and $C$ respectively. If $a=3, b=4$ and $A=\sin^{-1}\left(\frac{3}{4}\right)$,then the angle $B$ is (in $^{\circ}$)

If $4 \sin ^{-1} x + 6 \cos ^{-1} x = 3 \pi$,where $-1 \leq x \leq 1$,then $x =$

If the area of the region bounded by the parabola $y^2 = 4kx$ and the line $x = k$ (where $k > 0$) is $\frac{128}{3} \text{ sq. units}$, then the value of $\sin^{-1}(\frac{2}{k})$ is equal to...