Question

Prove the following identities:

`((sec θ - tan θ))/((sec θ + tan θ)) = (1 + 2 tan^2θ - 2 sec θ tan θ)`

Answer 1

LHS = `((sec θ - tan θ))/((sec θ + tan θ)) xx ((sec θ - tan θ))/((sec θ - tan θ))`

= `(sec θ - tan θ)^2/((sec^2θ - tan^2θ)`

= (sec θ – tan θ)²

= sec²θ + tan²θ – 2 sec θ tan θ   ...[∵ sec²θ = 1 + tan²θ]

= 1 + 2 tan²θ – 2 sec θ tan θ = RHS