Question

How do you verify the Identity: `(1+cos x+ sinx)/(1+cos x-sin x)=sec x+tan x`?

Answer 1

`(1+cosx+ sinx) / (1+cosx-sinx)=secx+tanx | * (1+cosx-sinx)`
`1+cosx+ sinx=(1/cosx+sinx/cosx)* (1+cosx-sinx)`
`=(1+cosx-sinx)/cosx+(1+cosx-sinx)*sinx/cosx`

`(1+cosx-sinx)/cosx=color(blue)(secx+1-tanx)`
`(1+cosx-sinx)*sinx/cosx=color(red)(tanx+sinx-sinxtanx)`

`1+cosx+ sinx=color(blue)(secx+1cancel(-tanx))+color(red)(cancel(tanx)+sinx-sinxtanx)`
`1+cosx+ sinx=secx+1+sinx-sinxtanx|-1-sinx`
`cosx=secx-sinxtanx|*cosx`
`cos^2x=1-sin^2x|+sin^2x`
`cos^2x+sin^2x=1`

Answer 2

`LHS=(1+cosx+ sinx) / (1+cosx-sinx)`

`=(cosx(1+cosx+ sinx) )/ (cosx(1+cosx-sinx))`

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

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

`=(cosx(1+sinx)+ (1-sinx)(1+sinx)) / (cosx(1+cosx-sinx))`

`=((1+sinx)(cosx+ 1-sinx)) / (cosx(1+cosx-sinx))`

`=(1+sinx)/cosx`

`=1/cosx+sinx/cosx`

`=secx+tanx=RHS`