I think there is an error in your problem the LHS should be +1 not -1
If you repeat the above mentioned steps, change the -1 into +1 in the RHS and +1 to -1 in the LHS, you should get the solution to yor problem.
I just realized that I originally posted the original equation wrong! The right one is the one that I just posted. Oops!
I'm pretty sure that the equation should look like this:
sec²x/secx-1 = csc²x(secx+1)
I think you switched the signs on each side of the equation, but I might not have written the equation down correctly. Thanks for your help!