How to sketch the spectrum of \$m(t) = \cos(1000)t \times \cos(3000)t\$
Maybe you remember the sine and cosine product formulas from high school?  

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.