Question

Use a double angle formula to show that cot(pi/12) satisfies the equation c^2 -2sqrt3c -1 =0 Deduce that cot(pi/12) = 2+sqrt3

Answer 1

\[\tan\left(\frac{\pi}{12}\right)=\tan\left(\frac{1}{2}\frac{\pi}{6}\right)\]
\[\tan\left(\frac{\pi}{12}+\frac{\pi}{12}\right)=(2\tan(\pi/12))/(1-\tan^2(\pi/12))\]
\[\tan\left(\frac{\pi}{12}+\frac{\pi}{12}\right)=\tan(\pi/6)=1/\sqrt{3}\]
Let \[c=1/\tan(\pi/12)\]Then
\[1/\sqrt{3}=(2/c)/(1-1/c^2)\]
\[c^2-2\sqrt{3}c-1=0\]
Solving the quadratic equation (choosing the plus sign, to stay in the first quadrant)
\[c=2+\sqrt{3}\]

Answer 2

Brilliant! Thank you so much