Hi,
Yes, the two formulas are not exactly equivalent, just correct as they are originally written which is what i looked at.
To help make this more confusing, i found another formula that is even different than those two and leads to a multiplication of ALL three terms by Vc*Vm.
I think what might be happening is each formula is assuming a (perhaps different) normalized parameter such as Vc. With Vc=1 the second formula works the same as the first formula.
The formula i tested was:
Vc*cos(wc*t)*Vm*(1+cos(wa*t))
where
wc is the carrier frequency and wa is the message frequency.
This led to a signal that definitely had sum and different frequencies present at 1/2 amplitudes and carrier present too.
I guess they like to put Vm inside the parens like so:
Vc*cos(wc*t)*(1+k*Vm*cos(wa*t))
and then state some limits for k.
What you could do is simulate both of your original signals and vary m or Vm and see what you can find out. I have a feeling they are both right but with different assumptions applied. If you have to take a test on this, you should ask the instructor about this too.