1+sin2x=(sinx)^2+(cosx)^2+2sinxcosx
=(sinx+cosx)^2
2(cosx)^2+sin2x=2cosx(cosx+sinx)
(1+sin2x)/[2(cosx)^2+sin2x]
=(sinx+cosx)^2/2cosx(cosx+sinx)
=(sinx+cosx)/2cosx
=sinx/2cosx + cosx/2cosx
=1/2tanx+1/2