Functions here and there

$Sin[2*Cos^{-1}\{Cot ( 2*Tan^{-1}X)\}]=0,\\ \implies ~2*Cos^{-1}\{Cot ( 2*Tan^{-1}X)\}=0,~~or~~2*\pi~~~OR~~~\pi.\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ............. 0\le {\Large \color{#3D99F6}{2} }*Cos^{-1}\{Cot ( 2*Tan^{-1}X)\} \le 2*\pi.\\ \implies ~Cos^{-1}\{Cot ( 2*Tan^{-1}X)\}=0,~~or~~\pi~~~OR~~~\dfrac{\pi}2.\\ \implies~ \color{#3D99F6}{ Cot ( 2*Tan^{-1}X)=1,~~or~~-1~~~OR~~~0}.\\ Let~~ \theta=2*Tan^{-1}X, ~~\implies X=Tan\dfrac \theta 2. \\ \therefore~Tan \theta=\dfrac {2X}{1 - X^2}. \\ \implies~ \color{#3D99F6}{ \dfrac {1 - X^2} {2X}=Cot \theta=Cot ( 2*Tan^{-1}X).}\\ \therefore~ \dfrac {1 - X^2} {2X}=\pm 1~~~OR~~~~~~~0.\\ Solving~~ the~~ quadratic,\\ n=6,\\ X_1=-1 +\sqrt2,~~~~~~~X_2= -1 - \sqrt2.\\ X_3= 1 +\sqrt2,~~~~~~~~~~X_4= 1 - \sqrt2 .\\ X_5=1 ,~~~~~~~~~~~~~~~~~~~~ X_6=- 1.\\ \left\{n + \sum\limits_{i=1}^n \left({\left|a_i\right| + \left|\sqrt{b_i}\right|}\right)\right\}\\ =6+2*|- 1|+ 2*|\pm \sqrt2|+2*|1|+ 2*|\pm \sqrt2|+2*|\pm 1|\\ = \Large \color{#D61F06}{ 17.657}$

Let's take the expression, $\sin\left[2\cos^{-1}\left\{\cot\left(2\tan^{-1}x\right)\right\}\right]=0$

For $\cos^{-1} x$ Principle domain is $\theta \in\left[0, \pi\right]\\\Rightarrow 2\theta \in [0, 2\pi]$

So, $\theta = 0, \dfrac{\pi}2, \pi$ . Applying $\cos$ , $\cot\left\{2\tan^{-1}x\right\} = 1, 0, -1$

OR $\dfrac{1-x^2}{2x}=1,0,-1$

Solving, $\Huge\boxed{ n=6\\\begin{aligned}x_1,~x_2&=-1\pm\sqrt 2\\x_3,~ x_4 &= \pm 1\\x_5,~x_6 &= 1 \pm\sqrt 2 \end{aligned}}$