Question

Ctg(1/2 arccos (-1/3))

Answer 1

$\displaystyle ctg(\frac{1}{2}arccos(- \frac{1}{3}))=ctg( \frac{arccos(-1/3)}{2})=\\\\= \frac{sin(arccos(-1/3))}{1-cos(arccos(-1/3)}=\frac{sin(\pi-arccos(1/3))}{1-cos(\pi- arccos(1/3))}= \frac{sin(arccos(1/3))}{1-(-cos(arccos(1/3))}=\\\\= \frac{sin(arccos(1/3))}{1+cos(arccos(1/3))}= \frac{\sqrt{1-(1/3)^2}}{1+1/3}= \frac{\sqrt{8/9}}{4/3}= \frac{2\sqrt{2}}{3}: \frac{4}{3}= \frac{\sqrt{2}}{2}$