I am currently trying to prove the stability of a numerical method applied to the PDE $$u_{t} = au_{xx} + 2bu_{xy} + cu_{yy}, $$ where $a, c >0$, and $ac \geq b^2$. I decided to use Crank-Nicolson, and I'm fairly confident that my analysis up until this point is correct, and should be irrelevant to my question. As the final step I need to prove the inequality \begin{align*}a\sin^2(\theta) + c\sin^2(\phi) + 2b\sin(\theta)\cos(\theta) \sin(\phi) \cos(\phi) \geq 0, \end{align*}
for all $\theta, \phi \in \mathbb{R}$, which would suggest that the method is unconditionally stable. I've tried plugging in various values of $a,b,c$ into the inequality and graphing the solution, and it seems to be true for all $\theta$ and $\phi$. Am I missing some obvious trig identity? Thanks in advance!
Hint: start with the inequality
$$a\sin^2(\theta) + c\sin^2(\phi) + 2\lambda\sqrt{ac}\sin(\theta)\sin(\phi)\ge0$$ valid for any real $|\lambda|\le1$.