How to Prove an Equality by Using Periodicity Identities
Using the periodicity identities comes in handy when you need to prove an equality that includes the expression (x + 2pi) or the addition (or subtraction) of the period. For example, to prove
follow these steps:

Replace all trig functions with the appropriate periodicity identity.
You’re left with (sec x – tan x)(csc x + 1).

Simplify the new expression.
For this example, the best place to start is to FOIL:
Now convert all terms to sines and cosines to get
Then find a common denominator and add the fractions:

Apply any other applicable identities.
You have a Pythagorean identity in the form of 1 – sin^{2} x, so replace it with cos^{2} x. Cancel one of the cosines in the numerator (because it’s squared) with the cosine in the denominator to get
Finally, this equation simplifies to cot x = cot x.