For $x > 0$, if $\sin(\cos^{-1}x + \tan^{-1}x) - \cos(\sin^{-1}x + \tan^{-1}x) = \sin(\cot^{-1}2)$ then $x = $

If $\tan^{-1}ax + \tan^{-1}3x = \dfrac{\pi}{4}$, where $3ax^2 < 1$, then value of $a$ for $x = \dfrac{1}{6}$ is $\ldots$

$\sec^2(\tan^{-1}3) - \tan^2(\sec^{-1}3) = $

If $y = \tan^{-1}\left(\dfrac{\log\left(\dfrac{e}{x^3}\right)}{\log ex^3}\right) + \tan^{-1}\left(\dfrac{\log(e^4x^3)}{\log\left(\dfrac{e}{x^{12}}\right)}\right)$, $x \in \left(e^{-\frac{1}{3}}, e^{\frac{1}{12}}\right)$ then $\dfrac{dy}{dx}$ is equal to...