The identity in your post is (I copy it)

    1 + sin(theta)/cot^2(theta) = sin(theta)/csc(theta) - 1.


I read it this way - exactly as it is written in your post

    1 +  =  - 1.



        This identity is INCORRECT, and I will show it right now.



Take  = 45°.

Then in the left side,  cot(theta) = cot(45°) = 1;  so,  cot^2(theta) = 1, and the left side is

    1 + sin(45°) = 1 + .


The right side is   - 1 =  - 1 =  - 1 =  - 1 =  - 1 = .


The left side 1 +  is not equal to the right side ,  so the identity is not hold.

What you wrote was NOT an identity.  You wrote this:

1 + sin theta/cot^2 theta = sin theta/csc theta - 1

But you should have written this:

(1 + sin theta)/cot^2 theta = sin theta/(csc theta - 1)

Without the proper placing of parentheses, your problem
cannot be solved. Be careful!!!

OK, you want to start with the right side:

  

If you want to know how I got any step, just ask in the thank-you note.

Edwin

I am using the fundamental identities to try to verify the following equation. It is a true statement but I find myself a little lost, probably on something basic algebra.

I primarily work on the left side and haven't really worked on the right side.

I know an identity could be verified by using many different methods, which is why the book I am using doesn't explain the process in the answers. So, your manner of verification may differ than what I am doing. Nevertheless, I'd appreciate any help.

In any case, here is the statement to be verified.

1 + sin theta/cot^2 theta = sin theta/csc theta - 1

Thank you very much for your help.

Sir Edwin manipulated the right side to match the left. Hence, I will do the same to the LEFT-SIDE to match the right side.

  

 ----- Substituting  for 

 ----- Factoring  in denominator

 --- Substituting