Prove 1-cosx+sinx / 1+cosx+sinx =tan x/2

Answer 1

use the properties
sin 2A = 2sinAcosA
and
cos 2A = cos^2 A - sin^2 A

let A = x/2

so LS = (1 - cosx + sinx)/(1 + cosx + sinx)
= (1 - cos 2A + sin 2A)/(1 + cos 2A + sin 2A)
= (cos^2 A + sin^2 A - (cos^2 A - sin^2 A) + 2sinAcosA) /(sin^2 A + cos^2 A + cos^2 A - sin^2 A + 2sinAcosA)
=(2sin^2 A + 2sinAcosA)/(2cos^2 A + 2sinAcosA)
= 2sinA(sinA + cosA)/(2cosA(cosA + sinA)
= 2sinA/(2cosA)
= sinA/cosA
= tanA
= tan (x/2)
= RS

Answer 2

∫ sin2xcos3x dx using factor formula