Question

Prove the following identities:

`(1 - sectheta + tan theta)/(1 + sec theta - tan theta) = (sectheta + tantheta - 1)/(sectheta + tantheta + 1)`

Answer 1

We know that,

1 + tan²θ = sec²θ

∴ sec²θ – tan²θ = 1

∴ (secθ – tanθ)(secθ + tanθ) = 1·1

∴ `(sectheta + tantheta)/1 = 1/(sectheta - tantheta)`

By the theorem of equal roots,

`(sectheta + tantheta)/1 = 1/(sectheta - tantheta)`

= `(sectheta + tantheta + 1)/(1 + sectheta - tantheta)`

= `(sectheta + tantheta - 1)/(1 - sectheta + tantheta)`

∴ `(sectheta + tantheta + 1)/(1 + sectheta - tantheta) = (sectheta + tantheta - 1)/(1 - sectheta + tantheta)`

∴ `(1 - sectheta + tantheta)/(1 + sectheta - tantheta) = (sectheta + tantheta - 1)/(sectheta + tantheta + 1)`.