The value of $\cot \left(\operatorname{cosec}^{-1} \frac{5}{3}+\tan ^{-1} \frac{2}{3}\right)$ is

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

Considering only the Principal values of inverse functions, the set

$$A=\left\{x \geq 0 \left\lvert\, \tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4}\right.\right\}$$

If $0< x < 1$, then

$$\sqrt{1+x^2}\left[\left\{x \cos \left(\cot ^{-1} x\right)+\sin \left(\cot ^{-1} x\right)\right\}^2-1\right]^{\frac{1}{2}}=$$