My question is
How to simplify $$f(x,y)=\delta(\cos x\cos y-k)\delta(\cos x\sin y)$$ ($0<k<1$) in terms of Dirac combs?
Background: I was working on a physics problem, in which I used $$g(\theta,\phi)=\lambda\cdot\delta(r\cos \theta\cos \phi-a)\delta(r\cos \theta\sin \phi)$$ (in spherical coordinates, $\lambda$ being the linear charge density) to represent a line of charge at $x=a,y=0$ in 3D Cartesian coordinates.
For some mathematical reasons, I need to find its Fourier transform, i.e. $$\int^\infty_{-\infty}\int^\infty_{-\infty}d\theta d\phi \cdot e^{-ik_1\theta-ik_2\phi}\cdot g(\theta,\phi)$$
I tried to work it out by brute force, and obtained $$\frac{4\lambda C(k_1)C(k_2)}{a\sqrt{r^2-a^2}}\cos\left(k_1\cos^{-1}\frac ar\right)$$ where $C(\cdot)$ is the $1$-periodic Dirac comb.
However, when I checked its correctness by performing an inverse Fourier transform on it, the original expression is not recovered.
Therefore, I would like to first simplify $g$ to reduce the chance of making errors in the second attempt. My ansatz is something proportional to $$C\left(\frac{\theta+\cos^{-1}\frac ar}{\pi}\right)C\left(\frac\phi\pi\right)$$ but I cannot proceed due to my lack of understanding in distribution theory.
Thanks in advance for any help.