Question

Simplify `(tanx+secx-1)/(tanx-secx+1)`?

Answer 1

Please see the proof below.

Explanation:

`(tanx+secx-1)/(tanx-secx+1)`

Multiplying numerator and denominator by `tanx+secx+1`

= `(tanx+secx-1)/(tanx-secx+1)xx(tanx+secx+1)/(tanx+secx+1)`

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

= `(tan^2x+2tanxsecx+sec^2x-1)/(tan^2x+2tanx-sec^2x+1)`

As `sec^2x=tan^2x+1`, above is equal to

= `(tan^2x+2tanxsecx+tan^2x+1-1)/(tan^2x+2tanx-tan^2x-1+1)`

= `(2tan^2x+2tanxsecx)/(2tanx)`

= `(2tanx(tanx+secx))/(2tanx)`

= `tanx+secx`