Question

How do you verify the identity `(tanx-secx)/(tanx+secx)=(tan^2x-1)/(sec^2x)`?

Answer 1

We have:

`(sinx/cosx - 1/cosx)/(sinx/cosx + 1/cosx) = (sin^2x/cos^2x- 1)/(1/cos^2x)`

`((sinx -1)/cosx)/((sinx +1)/cosx) = ((sin^2x - cos^2x)/cos^2x)/(1/cos^2x)`

`(sinx - 1)(sinx+ 1) = sin^2x - cos^2x`

`sin^2x - 2sinx - 1= sin^2x - cos^2x`

`sin^2x - 2sinx - (sin^2x + cos^2x) = sin^2x - cos^2x`

`-2sinx - cos^2x = sin^2x - cos^2x`

Since `-2sinx != sin^2x`, this identity is false.

Hopefully this helps!