Question

prove that sec theta - tan theta / sec theta + tan theta = 1 - 2 sec theta. tan theta+tan2theta​

Answer 1

To Prove:

secθ-tanθ/secθ+tanθ = 1-2 secθ.tanθ+tan²θ

Proof:

Rationalise LHS:

(secθ-tanθ)(secθ-tanθ)/(secθ+tanθ)(secθ-tanθ)

= (secθ-tanθ)²/(sec²θ-tan²θ)

= (secθ-tanθ)²/1                                            (As sec²θ-tan²θ=1)

= sec²θ+tan²θ-2 secθ.tanθ

= (1+tan²θ)+tan²θ-2 secθ.tanθ                     (As sec²θ=1+tan²θ)

= 1+2 tan²θ-2 secθ.tanθ

= 1-2 secθ.tanθ+2 tan²θ = RHS

Hence proved that RHS + LHS.