Question

Обчислити Ctg7pi/12

Answer 1

${\rm ctg}\,\dfrac{7\pi}{12}={\rm ctg}\,\left(\dfrac{\pi}{2}+\dfrac{\pi}{12}\right)=-{\rm tg}\,\dfrac{\pi}{12}=-{\rm tg}\,\left(\dfrac{\pi}{4}-\dfrac{\pi}{6}\right)=\\ \\ \\ =-\dfrac{{\rm tg}\,\dfrac{\pi}{4}-{\rm tg}\,\dfrac{\pi}{6}}{1+{\rm tg}\,\dfrac{\pi}{4}\cdot {\rm tg}\,\dfrac{\pi}{6}}=-\dfrac{1-\dfrac{1}{\sqrt{3}}}{1+\dfrac{1}{\sqrt{3}}}=\dfrac{1-\sqrt{3}}{\sqrt{3}+1}$