how to prove this curious identity with the Chebyshev polinomials

Question

we defined the Tm like this (where Tm are the Chebyshev polinomials)

Then I showed this:

And now I have no idea how to proove this:

I also have to make the remark that I also proved that the characteristic function of this matrix corresponds to the normed Chebyshev polinomials

It would very nice, if somebody could give me a hint. I tried induction but it didn't worked because the (m+1)-st Chebyshev polinomial has not the same roots as the m-th Chebyshev polinomial.