Question 1108163
{{{(1-cosx)/sinx = +- sqrt((cscx-cotx)/(cscx+cotx))}}}
(1-2cos+cos^2)/sin^2 = (csc-cot)/(csc+cot)
(1-2cos+cos^2)/sin^2 = (1-cos)/(1+cos)
(1-2cos+cos^2)/sin^2 = (1-cos)^2/(1-cos^2)
(1-2cos+cos^2)/sin^2 = (1-cos)^2/sin^2
(1-2cos+cos^2) = (1-cos)^2
QED
==================
You prove it working on it, not by asking about it.
I prefer to change to sine and cosine.
-----------
PS  I've never seen an example where working only one side makes a difference.