Adaboost - Show that adjusting weights brings error of current iteration to 0.5
You're nearly there.  The quantity $err_t/(1-err_t)$ is exactly what you need it to be.  It might be easier to see if you think about
$$\sum_{i=1}^N w_i \Pi(h_t(x^{(i)})=t^{(i)})$$
as 
$$\sum_{i:\ x^{(i)}\text{ is correctly classified}} w_i$$
(just using the indicator function to reduce the summation range).