Show `(tanx)/(secx+1)=(secx-1)/tanx` :

`(tanx)/(secx+1)`

`=tanx/(secx+1)*(secx-1)/(secx-1)` Multiplying by 1

`=(tanx(secx-1))/(sec^2x-1)`

`=(tanx(secx-1))/(tan^2x)` Pythagorean identity

`=(secx-1)/tanx` as required.

Show `(tanx)/(secx+1)=(secx-1)/tanx` :

`(tanx)/(secx+1)`

`=tanx/(secx+1)*(secx-1)/(secx-1)` Multiplying by 1

`=(tanx(secx-1))/(sec^2x-1)`

`=(tanx(secx-1))/(tan^2x)` Pythagorean identity

`=(secx-1)/tanx` as required.