When you integrate in spherical coordinates, the differential element isn't just $ d\theta d\phi $. No. It's $\sin\theta d\theta d\phi$, where $\theta$ is the inclination angle and $\phi$ is the azimuthal angle.
For example, attempting to integrate the unit sphere without the $\sin\theta$ term:
$$ \int_0^{2\pi}\int_0^{\pi} d\theta d\phi = 2 \pi^2 $$
With the $\sin\theta$ term you get
$$ \int_0^{2\pi}\int_0^{\pi} \sin\theta d\theta d\phi = 4 \pi $$
But I'm puzzled why you need to multiply the differential solid angle by $\sin\theta$. It would seem it's because the chunks at the north/south pole are "worth less" than the chunks at the equator. That kind of makes sense because they will be closer together.