  For the Hahn and Krawtchouk polynomials orthogonal on the set $\{0,
\ldots,N\}$ new identities for the sum of squares are derived which generalize
the trigonometric identity for the Chebyshev polynomials of the first and
second kind. These results are applied in order to obtain conditions (on the
degree of the polynomials) such that the polynomials are bounded (on the
interval $[0,N]$) by their values at the points $0$ and $N$. As special cases
we obtain a discrete analogue of the trigonometric identity and bounds for the
discrete Chebyshev polynomials of the first and second kind.
