#Eq_1: cot theta = 1/tan theta = cos theta/sin theta#
On the equation:
#sin theta + cos theta cot theta = #
Now, substitute the #cot theta# for the value we found on #Eq_1#:
#sin theta + cos theta * cos theta/sin theta #
Apply the multiplication:
#sin theta + (cos²theta)/sin theta#
To purpose of simplification the whole Equation must have the same denominator. In order to do that, multiply #sin theta# by #sin theta/sin theta#, that is 1. Therefore, we are not making changes on equality (since anything multiplied by 1 results on itself):
#sin theta*(sin theta)/sin theta + (cos²theta)/sin theta = (sin²theta)/sin theta + (cos²theta)/sin theta = (sin² theta + cos² theta)/sin theta = 1/sin theta = csc theta#
