= Trigonometric type-two canonical transformation
{title2=$F_2(q,P)=\cos q\sin P$}
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 $\cos P\ne0$, its inverse in mixed coordinates is
$$
q=\arccos(Q/\cos P),\qquad
p=-\sin P\sqrt{1-Q^2\sec^2P}.
$$
The square root equals $\sin q$ on this branch. Differentiating $F_2$ proves the generating equations, and $F_{2,qP}=-\sin q\cos P\ne0$ proves local invertibility by the <inverse function theorem>. Also $d(F_2-PQ)=p\,dq-P\,dQ$, 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.
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