Both sin and cos are periodic with period 2pi.
Then, for example, sin(t)+cos(t) is automatically periodic of 2pi because if we substitute t=2pi both functions return on the initial value and so does their sum.
Now the period of the function sin(t/32) is 64pi because when t=64pi we have sin(2pi) that is equal to sin(0) and then the function restarts.
Applying the same concept cos(t/64) has the period 128pi.
This means that if we take the sum, when we arrive to 64pi the sin did a full turn but the cos is still not repeating. When we are at 128pi the sin did two full turns (4pi) and the cos did its full period. So both functions are again to zero and the sum will restart the next cycle.
We are lucky that 128 is exactly the double of 64 so one period of the cos correspond to exactly two periods of sin. If this is not true we have to search the least common multiple of both periods to have a period that is valid for both functions. In fact 128 is the LCM of 128 and 64.