Question

Prove that `(1 + sec θ - tan θ)/(1 + sec θ + tan θ) = (1 - sin θ)/(cos θ)`.

Answer 1

sec² θ − tan² θ = 1

(sec θ − tan θ) (sec θ + tan θ) = 1

L.H.S = `((sec θ - tan θ) (sec θ + tan θ) + (sec θ - tan θ))/(1 + sec θ + tan θ)`

= `((sec θ - tan θ) [sec θ + tan θ + 1])/ ([1 + sec θ + tan θ])`

= sec θ − tan θ

= `1/cos θ - sin θ/cos θ = (1 - sin θ)/cos θ`

L.H.S = R.H.S

Hence Proved.