<s>[INST] The value of the expression $x^4-8x^3+18x^2-8x+2$ when $x=\cot\frac{\pi}{12}$ The value of the expression $x^4-8x^3+18x^2-8x+2$ when $x=\cot\frac{\pi}{12}$

I know that $\cot\frac{\pi}{12}=2+\sqrt3$,but putting this value and evaluating is a long process.Isn't there a short and intuitive method to solve this?
 [/INST] Hint:
$$\cot15^\circ+\tan15^\circ=\dfrac2{\sin(2\cdot15^\circ)}=?$$
$$\cot15^\circ\tan15^\circ=?$$
So, $\cot15^\circ,\tan15^\circ$ are the roots of $$t^2-4t+1=0$$
Now we can straightaway divide $$x^4-8x^3+18x^2-8x+2$$ by $$x^2-4x+1$$ to find the remainder.
</s>