Question

求证 (sinθ+cosθ-1)(sinθ-cosθ+1)) /sin2θ=tanθ/2

Answer 1

[(sinθ+cosθ-1)(sinθ-cosθ+1)]/sin2θ

=[(sinθ)^2-(cosθ-1)^2]/(2sinθcosθ)

=[(sinθ)^2-(cosθ)^2+2cosθ-1]/(2sinθcosθ)

=[2cosθ-2(cosθ)^2]/(2sinθcosθ)

=(1-cosθ)/sinθ

=[1-(1-2(sinθ/2)^2)]/(2sinθ/2cosθ/2)

=[2(sinθ/2)^2]/(2sinθ/2cosθ/2)

=sinθ/2/cosθ/2

=tanθ/2