Final answer:
The function cos(x) cos(ax) is periodic when a is a rational number because both component cosine functions will repeat at regular intervals, the intervals being integer multiples of their periods.
Step-by-step explanation:
To show that cos(x) cos(ax) is periodic when a is rational, first understand that a function is periodic if it repeats values at regular intervals, which is determined by its period. For cosine functions, the standard period is 2π. If a is a rational number, then it can be expressed as a fraction, say m/n, where m and n are integers. Therefore, ax could be written as (m/n)x. Now consider the two functions cos(x) and cos((m/n)x). The period of cos(x) is 2π. For cos((m/n)x), the period will be the value such that (m/n)x completes one full cycle, which would be 2nπ/m. After this interval, both cosines will complete integer multiples of their periods, and thus the product cos(x) cos((m/n)x) will repeat.