Calculator Thug Life Compilation!

Let $x=0.5$ . Then, $\cot x - \tan x = \frac{\cos x}{\sin x}-\frac{\sin x}{\cos x} = \frac{2\left(\cos ^2x-\sin ^2x\right)}{2\sin x\cos x} = \frac{2\cos 2x}{\sin 2x} = 2\cot 2x$

Now,

$A = \cot x - \tan x - 2 \tan 2x - 4 \tan 4x - 8 \cot 8x$

$= 2\cot 2x - 2 \tan 2x - 4 \tan 4x - 8 \cot 8x$

$= 2(\cot 2x - \tan 2x) - 4 \tan 4x - 8 \cot 8x$

$= 2(2 \cot 4x) - 4 \tan 4x - 8 \cot 8x$

$= 4( \cot 4x - \tan 4x) - 8 \cot 8x$

$= 4(2 \cot 8x) - 8 \cot 8x$

$= 8 \cot 8x - 8 \cot 8x = 0$

Hence, $A+20=0+20=\boxed{20}$

Let $x = 0.5^\circ$ . Then we have:

$\begin{aligned} A & = \color{#3D99F6}{\cot x - \tan x} - 2 \tan 2x - 4 \tan 4x - 8 \cot 8x \\ & = \color{#3D99F6}{\frac 1{\tan x} - \tan x} - 2 \tan 2x - 4 \tan 4x - \frac 8{\tan 8x} \\ & = \color{#3D99F6}{\frac {1-\tan^2 x}{\tan x}} - 2 \tan 2x - 4 \tan 4x - 8 \cot 8x \\ & = \color{#3D99F6}{2\cot 2x} - 2 \tan 2x - 4 \tan 4x - 8 \cot 8x & \small \color{#3D99F6}{\implies \cot \theta - \tan \theta = 2 \cot 2 \theta} \\ & = 4 \cot 4x - 4 \tan 4x - 8 \cot 8x \\ & = 8 \cot 8x - 8 \cot 8x \\ & = 0 \end{aligned}$

$\implies A + 20 = 0 + 20 = \boxed{20}$