simplifying expressions – Calculate density of states for 1D system


The density of states for a 1D system can be written as:

$Dleft(omegaright)=frac{L}{pi}frac{1}{domegaleft(kright)/dk}$

I have some expressions for $omegaleft(kright)$:

w1(k_) = Sqrt((k1 (m1 + m2)/(m1*m2))*(1 - 
      Sqrt(1 - (2*(1 - Cos(k*a)) m1*m2)/((m1 + m2)^2))));
w2(k_) = Sqrt((k1 (m1 + m2)/(m1*m2))*(1 + 
      Sqrt(1 - (2*(1 - Cos(k*a)) m1*m2)/((m1 + m2)^2))));

I need to calculate $Dleft(omegaright)$ analytically eliminating $k$ but I can only do that if I simplify the expression for w1,2(k) by assigning values to k1,m1,m2,a:

k1 = 1; a = 1; m1 = 1; m2 = 2;
w1(k_) = Sqrt((k1 (m1 + m2)/(m1*m2))*(1 - 
      Sqrt(1 - (2*(1 - Cos(k*a)) m1*m2)/((m1 + m2)^2))));
w2(k_) = Sqrt((k1 (m1 + m2)/(m1*m2))*(1 + 
      Sqrt(1 - (2*(1 - Cos(k*a)) m1*m2)/((m1 + m2)^2))));
sol = Solve(p == D(w2(k), k) && w == w2(k), {p}, {k});
Simplify((p /. sol((2)))^-1)

Output:

(2 Sqrt((-3 + 2 w^2)^2))/Sqrt(6 - 11 w^2 + 6 w^4 - w^6)

Is it possible to do it for the general case?