Problem: Determine periodicity of the following function:

Solution: We always start by checking on the domain. Here there are two problems. First, the tangent rules out all numbers of the form π/2 + kπ. We also have a fraction, the denominator cannot be zero. Thus the cosine cannot be −1, therefore numbers (2x) cannot be (2k + 1)π, that is, x cannot be π/2 + kπ. Remarkably enough, we've got the same condition from tangent and cosine, but that's just a happy coincidence. Thus the domain is all real numbers apart from the type listed here.

Now back to the periodicity. The function consists of three composed functions combined together using algebraic operations. First we look at each composed function.

sin(3x) has periodicity (2π)/3, since it is a 2π-periodic sine whose argument was scaled (see Methods Survey - Real functions - Transformations).

The second function we look at is cos(2x), which is - by the same argument - π-periodic. We can even include the "+ 1" part, we know that shifting a function up does not change its period.

The third term is tan³(x). Tangent is π-periodic and cubing it does not change the period (when composing functions and the inner one is periodic, the outer one does not spoil it, again see Methods Survey - Real functions - Transformations).

So we have three functions whose periodicity we know and they are combined using division and addition. The rule says that in such a situation, the outcome has period which is the least common multiple of the periods of the individual parts. In our case we have periods 2π/3 and π, so we are looking for a positive real number r such that r/(2π/3) is an integer and r/π is an integer. The second condition says that r = kπ for some positive integer k, and the smallest k that would work for the first condition is 2.

We therefore claim that T = 2π is a period of the given function.

Confirmation:

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