Question

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

Answer 1

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