TRIGONOMETRY
Author: Jelly
(i) Prove that (cosecθ − cotθ)² = (1 − cosθ)/(1 + cosθ)
LHS = (cosecθ − cotθ)²
= (1/sinθ − cosθ/sinθ)²
= ((1 − cosθ)/sinθ)²
= (1 − cosθ)² / sin²θ
= (1 − cosθ)² / (1 − cos²θ)
= (1 − cosθ)² / [(1 − cosθ)(1 + cosθ)]
= (1 − cosθ)/(1 + cosθ) = RHS
(ii) Prove that cosθ/(1 + sinθ) + (1 + sinθ)/cosθ = 2secθ
LHS = [cos²θ + (1 + sinθ)²] / [(1 + sinθ)cosθ]
= [cos²θ + 1 + sin²θ + 2sinθ] / [(1 + sinθ)cosθ]
= [1 + 1 + 2sinθ] / [(1 + sinθ)cosθ]
= 2(1 + sinθ)/[(1 + sinθ)cosθ]
= 2/cosθ
= 2secθ = RHS
(iii) Prove that tanθ/(1 − cotθ) + cotθ/(1 − tanθ) = 1 + secθcosecθ
tanθ = sinθ/cosθ, cotθ = cosθ/sinθ
LHS = sin²θ/[cosθ(sinθ − cosθ)] − cos²θ/[sinθ(sinθ − cosθ)]
= (sin³θ − cos³θ)/[sinθcosθ(sinθ − cosθ)]
= (sinθ − cosθ)(sin²θ + cos²θ + sinθcosθ)/[sinθcosθ(sinθ − cosθ)]
= (1 + sinθcosθ)/[sinθcosθ]
= 1/(sinθcosθ) + 1
= secθcosecθ + 1
= 1 + secθcosecθ = RHS
