Hamiltonian, generating function, canonical transformation

[fluidistic]

Homework Statement

Consider a harmonic oscillator with generalized coordinates q and p with a frequency omega and mass m.
Let the transformation (p,q) -> (Q,P) be such that $F_2(q,P,t)=\frac{qP}{\cos \theta }-\frac{m\omega }{2}(q^2+P^2)\tan \theta$.
1)Find $K(Q,P)$ where $\theta$ is a function of time and K is the Hamiltonian in function of Q and P.
2)For which values of $\theta (t)$ does $K(Q,P)$ vanishes?

Homework Equations

$H=\frac{p^2}{2m}+\frac{\omega m }{2}$.
$p=\frac{\partial F_2}{\partial q}, Q=\frac{\partial F_2}{\partial P}$
$K=H+\frac{\partial F_2}{\partial t}$

The Attempt at a Solution

I guess the main idea is to calculate $\frac{\partial F_2}{\partial t}$ and express p and q in terms of P and Q.
Playing with the relevant equations I get that $q=Q\cos \theta +m \omega P \sin \theta$, $p=\frac{P}{\cos \theta }-m\omega \tan \theta (Q\cos \theta + m\omega P \sin \theta )$.
Also, $\frac{\partial F_2}{\partial t}=qP\dot \theta \sin \theta - \frac{m\omega }{2}(q^2+P^2)\left ( \frac{\dot \theta }{\sin ^2 \theta } \right )$.
This gave me $K=\frac{1}{2m} \left [ \frac{P}{\cos \theta }-m\omega (Q\cos \theta+m \omega \sin \theta ) \tan \theta \right ] ^2+\frac{m\omega}{2}(Q\cos \theta + m \omega P \sin \theta )^2+ (Q \cos \theta + m \omega P \sin \theta )P\dot \theta \sin \theta -\frac{m\omega }{2} [(Q\cos \theta + m\omega P \sin \theta)^2+P^2]\left ( \frac{\dot \theta }{\sin ^2 \theta } \right )$.
The new Hamiltonian does indeed depends on the variables it should, but it looks so horrible that I cannot believe I made things right. What am I missing?

 

[member 553083]

(The answer should be K=\frac{P^2}{2m}+\frac{m \omega ^2 Q^2}{2}).Also, I have no idea how to tackle the second part of the problem. Any help would be appreciated. Thank you!