Use one of the tan half-angle identity:
tan(theta/2)=+-sqrt((1-costheta)/(1+costheta))
color(white)(tan(theta/2))=(1-costheta)/sintheta
color(white)(tan(theta/2))=(sintheta)/(costheta+1)
I will use the third one, because it works out well in this case. First, factor out the tan(t/2), then use the identity, and lastly rewrite cost as 1/(1/cost):
LHS=tan (t/2) (cos^2t) - tan(t/2)
color(white)(LHS)=tan(t/2)(cos^2t-1)
color(white)(LHS)=tan(t/2)(cost-1)(cost+1)
color(white)(LHS)=sint/(cost+1)*(cost-1)(cost+1)
color(white)(LHS)=sint/color(red)cancelcolor(black)((cost+1))*(cost-1)color(red)cancelcolor(black)((cost+1))
color(white)(LHS)=sint*(cost-1)
color(white)(LHS)=sintcost-sint
color(white)(LHS)=sint*1/(1/cost)-sint
color(white)(LHS)=sint*1/(sect)-sint
color(white)(LHS)=sint/(sect)-sint
color(white)(LHS)=RHS
That's the proof. Hope this helped!