I have 2 different functions whose derivative is same!! explain why

1) (tanx - 1)/2(tanx + 1) w.r.t. x
and
2) (-1)/(tanx + 1) w.r.t. x


both has derivative 1/(cosx + sinx)^2

That is correct, because these two functions DIFFER BY A CONSTANT...

This is true for ANY pair of functions which have the same derivative.. They MUST differ by a constant.

To prove that (tanx - 1)/2(tanx + 1) and (-1)/(tanx + 1) differ by a constant, consider subtracting the SECOND function from the FIRST function = (tanx - 1)/2(tanx + 1) - (-1)/(tanx + 1)... We will show that this is some fixed constant.

= (tanx - 1) / 2(tanx + 1) + (1) / (tanx + 1)
=(tanx - 1) / 2(tanx + 1) + (2) / 2(tanx + 1)
= (tanx - 1 + 2) / 2(tanx + 1)
= (tanx + 1) / 2* (tanx + 1)
= 1/2

proving our claim.