# 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*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 getFinally, this equation simplifies to cot

*x*= cot*x.*