 Trigonometric type-two canonical transformation

The type-two generating function for a canonical transformation $F_2(q,P)=\cos q\sin P$ defines $p=-\sin q\sin P$ and $Q=\cos q\cos P$. On any branch with $0<q<\pi$ and $\cos P\ne 0$, its inverse in mixed coordinates isThe square root equals $\sin q$ on this branch. Differentiating $F_2$ proves the generating equations, and $F_{2,qP}=-\sin q\cos P\ne 0$ proves local invertibility by the inverse function theorem. Also $d(F_2-PQ)=pdq-PdQ$, so the symplectic form obeys $dq\wedge dp=dQ\wedge dP$. This proves the transformation is canonical locally. It is not a single global chart across the inverse-trigonometric branch boundaries.

Since $F_2$ is independent of time, the transformed Hamiltonian is obtained by substitution, with zero generating-function time correction. An added function of time alone is an irrelevant Hamiltonian freedom, since it leaves Hamilton's equations unchanged.