Steps to Solve

1. Start with the Left-Hand Side (L.H.S) The expression to transform is:

   $$ \text{L.H.S} = \sin(2A) - \sin(2B) + \sin(2C) $$

2. Use the Sine Difference Formula Apply the sine difference identity:

   $$ \sin(2A) - \sin(2B) = 2 \cos(A + B) \sin(A - B) $$

   Thus, we rewrite the L.H.S as:

   $$ 2 \cos(A + B) \sin(A - B) + \sin(2C) $$

3. Apply the Sine Double Angle Formula for $\sin(2C)$ Next, rewrite $\sin(2C)$ using the double angle formula:

   $$ \sin(2C) = 2 \sin(C) \cos(C) $$

   Now the expression becomes:

   $$ 2 \cos(A + B) \sin(A - B) + 2 \sin(C) \cos(C) $$

4. Factor out the common factor of 2 Factor out the common factor from the terms:

   $$ 2 \left(\cos(A + B) \sin(A - B) + \sin(C) \cos(C)\right) $$

5. Use sine sum and difference identities Replace the sine and cosine functions using the sine addition formula:

   Using $\sin(A + B) = \sin A \cos B + \cos A \sin B$ and $\sin(A - B) = \sin A \cos B - \cos A \sin B$.

   The expression can be transformed as follows:

   $$ 2 \csc[-\sin(A - B) + \sin(A + B)] $$

6. Further simplifications Continue simplifying and rearranging until reaching the desired identity. Apply the fact that $\sin(A+B) = \sin(\pi - (A+B))$.

7. Final expression Eventually, it resolves to a more simplified equivalent form depending on the specific identities used throughout the derivation.