Superposition of two waves with different frequencies

[SinSinger]

Homework Statement

Hi all! It's a superposition question: Two waves travel through dispersive medium, with different frequencies and wave number.
P1(t)=Acos(k1x-w1t)
P2(t)=Acos(k2x-w2t)
Obtain the P(t)=P1(t)+P2(t)

Homework Equations

Well I used identity:
cosα+cosβ=2 cos 1/2(α+β)cos1/2(α-β)
and the following:
w(av)=(w1+w2)/2 Δw=w1-w2
k(av)= (k1+k2)/2 Δk=K1-k2

The Attempt at a Solution

So, this is what I tried to do:
P(t)=A0(2cos(1/2)((k1x-w1t)+(k2x+w2t))cos(1/2)((k1x-w1t)-(k2x-w2t)

=2A0(cos(((k+k)/2)x)-((w-w)/2)t))cos(((k-k)/2)x)-((w+w)/2)t))

=2A0(cos(k(av)-(1/2)Δwt)cos((1/2)Δkx-w(av)t))

And, from here on I'm stuck: Is this all that needed? Help would be very appreciated. :)

 

[BvU]

I think this is all that's needed. But you have to be a bit sharper with the signs, to get k(av) and w(av) in the first and Δk and Δw in the second cosine - I think...

 

[SinSinger]

Hmm, what do you mean?
Thank for the help by the way :)