Question

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

Answer 1

LHS = `1/(sec θ - tan θ)`

= `1/((1/cos θ) - (sin θ/cos θ))`

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

= `(cos θ( 1 + sin θ))/(1 - sin^2 θ)`

= `(cos θ( 1 + sin θ))/(cos^2 θ)`

= `1/cos θ + sin θ/cos θ`

= sec θ + tan θ

= RHS

Hence proved.