# How to sketch the spectrum of $m(t) = \cos(1000)t \times \cos(3000)t$

I know how to sketch the spectrum of $m(t) = \cos(1000)t + \cos (3000)t$

But I stack with that....when it's multiplied? How to sketch the spectrum $m(t) = \cos(1000)t \times \cos(3000)t$?

Replace $a$ and $b$ with $1000t$ and $3000t$ and you get a sum of new frequencies. This is also the concept used in superheterodyne radio receivers.
If you know how to add two cosines, you can use the identity $$\cos(\alpha)\times\cos(\beta) = \frac{1}{2}\left[\cos(\alpha + \beta) + \cos(\alpha − \beta)\right]$$