Define a canonical transformation for a one-dimensional system with coordinates $(q, p) \rightarrow(Q, P)$. Show that if the transformation is canonical then $\{Q, P\}=1$.

Find the values of constants $\alpha$ and $\beta$ such that the following transformations are canonical: (i) $Q=p q^{\beta}, P=\alpha q^{-1}$. (ii) $Q=q^{\alpha} \cos (\beta p), P=q^{\alpha} \sin (\beta p)$.