Question

Prove the following identity : 

`(tanθ + 1/cosθ)^2 + (tanθ - 1/cosθ)^2 = 2((1 + sin^2θ)/(1 - sin^2θ))`

Answer 1

`(tanθ + 1/cosθ)^2 + (tanθ - 1/cosθ)^2`

= `(sinθ/cosθ + 1/cosθ)^2 + (sinθ/cosθ - 1/cosθ)^2`

= `((sinθ + 1)/cosθ)^2 + ((sinθ - 1)/cosθ)^2`

= `(sinθ + 1)^2/(cos^2θ)  + (sinθ - 1)^2/cos^2θ`

= `((sinθ + 1)^2 + (sinθ - 1)^2)/cos^2A`

= `(sin^2θ + 1 + 2sinθ + sin^2θ + 1 - 2sinθ)/(1 - sin^2θ)`

= `(2(1 + sin^2θ))/(1 - sin^2θ)`