Is $f(t-N t_0)=\sin(2\pi f_0t)\cos(2\pi f_1t)$ always true?

Question

Is it true that multiplying two sinusoidal functions, always result in some periodic waveform.

i-e $$f(t-N t_0)=\sin(2\pi f_0t) \cos(2\pi f_1t)$$

If so, then how can we calculate the period ( $t_0$) of the resulting function, f(t) ?

Moreover, is it correct to say that:

$$\int_{-\infty}^{\infty}\sin(2 \pi f_1 t)\sin(2 \pi f_2 t)\ dt= \delta(f_1-f_2)$$

thanks

Answer 1

1. In general no, but it is often a yes. You have the following formula: $$ \sin(2\pi f_0 t + \phi) \sin(2\pi f_1 t) =\frac 12\left[ \cos(2\pi (f_0-f_1) t + \phi) + \cos(2\pi (f_0+f_1) t + \phi) \right] $$ It is not periodic if the ratio of frequencies of both signals is not a rational number.
2. The integral you talk about is not well defined from the mathematical point of view of Lebesgue integral. It may be true in terms of distributions, I need a more precise statement (you can't define a product of distributions, as far as I know).