I do not know if i am going to help you, but:
Why dont you try to expand the identity as follow:
From binomial expansion we have:
And then we have the equality:
From there i have nothing more...
If we rise both sides to a expand the right side and the again rise to a,
mayby we could compare the addition terms.
if we try to simplify left side by divisions of 2.
again i do not know if this helps to prove (or disprove) your identity.