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.