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:

1. Replace all trig functions with the appropriate periodicity identity.

   You're left with (sec x – tan x)(csc x + 1).

2. 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:

   

3. Apply any other applicable identities.

   You have a Pythagorean identity in the form of 1 – sin² x, so replace it with cos² 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.