A more elegant solution for proving these trig. identities?

Hi there. I just learnt trigonometry today, and the teacher didn't really delve much into the topic yet, just the basic identities. I can prove these identities using normal maths rules, but not using the LHS /RHS rule as compared to what the teacher has taught us. For example;

Prove:

cosec x - cot x = (sin x)/(1 + cos x) ----- (1)

cosec x - cot x = 1/(sin x) - (cos x)/(sin x)

= (1 - cos x)/(sin x) ------(2)

Since (1)=(2),

(sin x)/(1 + cos x) = (1 - cos x)/(sin x)

sin^2 x= 1 - cos^2 x [Basic Identity]

That is what I do to these sort of identites involving linear equations. I have no prob. doing say (sin x)/(cos x - sin x) = cot x + 1

So, can anyone post some steps for proving this? Or is my method the only way? I'm sure there's a more elegant sol.

**(tan x + sec x - 1)/(tan x - sec x +1) = tan x +sec x**