3 cos^2 𝛼 + 2 cos^2 𝛽 =4

I already did this for you. Take the trouble to read it this time, ok?

since sin^2 + cos^2 = 1, the 2nd equation becomes
3(1-cos^2 𝛼) − 2(1-cos^2 𝛽) = 0
3 - 3cos^2 𝛼 − 2 + 2cos^2 𝛽 = 0
Now the two equations read
3cos^2 𝛼 + 2cos^2 𝛽 = 4
-3cos^2 𝛼 + 2cos^2 𝛽 = -1
subtract to get
6cos^2𝛼 = 5
cos^2𝛼 = 5/6
Now just finish solving for cos^2𝛼 and cos^2𝛽
Then use your double-angle formula
cos 2x = 2cos^2x - 1

post your work if you get stuck, but don't just post the same problem all over again.