Using the theory of Pell equations and the fact that the discriminant $5$ has both the forms $k^2 + 4$ and $k^2 - 4$, I stumbled across a proof of the identities
$$ (-1)^n T_{2n} (\tfrac{i}{2}) = T_n (\tfrac{3}{2}) $$ $$ (-i)^{2n+1} U_{2n+1} (\tfrac{i}{2}) = U_n (\tfrac{3}{2})$$
for all positive integers $n$. Here, $i$ is a square root of $-1$ and $T_n$ and $U_n$ denote Chebyshev polynomials of the first and second kinds.
My question is, are there general identities involving Chebyshev polynomials that specialize to these when appropriate inputs are substituted?
Partial quick answer: The Wolfram function site http://functions.wolfram.com/05.04.16.0006.01 lists the relation $$T_{2n}(z) = (-1)^n T_n(1 - 2 z^2)$$
If you substitute $z=\frac{i}{2}$ you get $$T_{2n}\left(\frac{i}{2}\right) = (-1)^n T_n\left(1 - 2 (\frac{i}{2})^2\right)= (-1)^n T_n\left(\frac{3}{2}\right)$$ The corresponding entry for $U$ (http://functions.wolfram.com/05.05.16.0004.01) needs some more work than direct substitution (maybe with recurrence formula).