An excellent teacher from my past said that verifying a Trig Identity was frequently a little bit of trig and a LOT of Algebra.
To verify or prove the trigonometric identity that follows, one can do the following steps:
1. Rewrite the left side, which is the sum of two reciprocal trig functions ( cscx and cotx ) ,in terms of sinx and cosx.
2. The preceding step will give you a common denominator.
3. Multiply the resulting fraction top and bottom by sinx.
4. The preceding step is confusing but justified by the fact that sin^2x = 1-cos^2x which is a pythagorean Identity.
5. Now we use the difference of two squares factoring formula in the next to last step to get a cancellation of a common factor in numerator and denominator which ends up verifying the identity.