prove that:tan theta + sec theta - 1 / tan theta -sec theta +1 = 1+sin theta /cos theta - Maths - Introduction to Trigonometry

Question

prove that:tan theta + sec theta - 1 / tan theta -sec theta +1 =
1+sin theta /cos theta

Priyanka Kedia · Accepted Answer

L H S
=tanθ+secθ-1tanθ-secθ+1          =sinθcosθ+1cosθ-1sinθcosθ-1cosθ+1          =sinθ+1-cosθsinθ-1+cosθ          =2sinθ2cosθ2+1-1+2sin2θ22sinθ2cosθ2-1+1-2sin2θ2          =2sinθ2cosθ2+sinθ22sinθ2cosθ2-sinθ2          =cosθ2+sinθ2cosθ2-sinθ2×cosθ2+sinθ2cosθ2+sinθ2          =cosθ2+sinθ22cos2θ2-sin2θ2          =cos2θ2+sin2θ2+2cosθ2+sinθ2cos2θ2-sin2θ2          =1+sinθcosθ          =R
H S Hence Proved