  In this paper, we determine the first cohomology of the monochromatic
comodule $M^1_2$ at an odd prime, and apply the results to show
non-trivialities of some products of beta and gamma elements in the homotopy
groups of the Smith-Toda spectrum $V(1)$. The cohomology for a prime greater
than three was determined by the first author. Here, we verify them and
determine the cohomology at the prime 3 by elementary calculation. The
cohomology will be a stepping stone for computing the cohomology of the
monochromatic comodule $M^3_0$, which we hope to determine for a long time.
