First thing with trig proofs, you need to decide which side to work with, its best to limit your self to solving one side.
I would start with factoring out (seca-tana)^2
I think what you mean is that the answer in the back of your book has
(1-sina)/(1+sina)= (1-sina)/(1+sina) as the proof for the identities.
If that is the case start with the left side of the equation (seca-tana)^2 and use your identities to work towards the (1-sina)/(1+sina).
Soroban just proved the identity by working on the right side first, I'm sure your instructor will tell you that either method is a valid solution.