How do you use the half angle identity to find cos 105?

Half Angle Formula

##cos(\theta/2)=pm sqrt{{1+cos theta}/2}##

First, since ##105^circ## is the 2nd quadrant, cosine is negative, so by the half angle formula above,

##cos(105^circ)=cos({210^circ}/2)=-sqrt{{1+cos(210^circ)}/2}##

by ##cos(210^circ)=-sqrt{3}/2##,

##=-sqrt{{1-sqrt{3}/2}/2}=-sqrt{{2-sqrt{3}}/4}=-{sqrt{2-sqrt{3}]}/2##

I hope that this was helpful.