Integral involving delta functions and vector quantities This integral comes from equation (3.15) in an older paper I've been reading:
$$
\int \mathrm{d} \Omega_k \, 
\delta\left(|\vec{k}|^2 - |\vec{k}+\vec{q}_1|^2\right)
\delta\left(|\vec{k}|^2 - |\vec{k}-\vec{q}_2|^2\right) .
$$
The integration over $\Omega_k$ refers to the angular coordinates of the vector $\vec{k}$. I'm most interested in the case of three spatial dimensions, for which
$$
\int \mathrm{d} \Omega_k 
\to
\int_0^{2\pi} d\theta \int_0^\pi d\varphi \sin \varphi,
$$
where $\theta$ and $\varphi$ are the azimuthal and polar angles associated with $\vec{k}$. The result should depend on the lengths $|\vec{k}|$, $|\vec{q}_1|$, and $|\vec{q}_2|$, as well as the dot product $\vec{q}_1\cdot\vec{q}_2$. 
The authors write that the integral is straightforward and don't give an explicit result, so I suspect that I'm missing something simple. I've tried a few strategies, such as using integral representations for the delta functions, but I haven't yet found an elegant approach.
 A: This isn't a full solution (yet), but it's the closest I've come so far. I still think there is likely a more elegant approach. I will use the convention that $k = |\vec{k}|$.
Coordinate system (three dimensions)
Choose a spherical coordinate system for $\vec{k}$ such that $\vec{q}_1$ coincides with the positive $z$-axis, and let $\theta$ and $\varphi$ represent the azimuthal and polar angles associated with $\vec{k}$. Then, 
$$
\vec{k} \cdot \vec{q}_1 = k \, q_1 \cos \varphi .
$$
Rotate the coordinate system about the z-axis until $\vec{q}_2$ lies on the $x>0$ portion of the $x$-$z$ plane. Let $\alpha$ represent the angle between $\vec{q}_2$ and the $z$-axis. Then, using the formula for a dot product in spherical coordinates,
$$
\vec{k} \cdot \vec{q}_2 = k \, q_2
\left[ \sin \varphi \sin \alpha \cos \theta + \cos \varphi \cos \alpha \right].
$$
The original integral may now be written
$$
\int_0^{2\pi} d\theta \int_0^\pi d\varphi \, \sin \varphi \,\,
\delta\left(f(\varphi)\right) \delta\left(g(\theta,\varphi)\right)
$$
with
$$
f(\varphi) = 2k \, q_1\cos\varphi + q_1^2
$$
and
$$
g(\theta,\varphi) = 2kq_2
\left[ \sin \varphi \sin \alpha \cos \theta + \cos \varphi \cos \alpha \right]
-q_2^2.
$$
Evaluating the integral over $\varphi$
Using the formula suggested by Calvin Khor, we can say that
$$
\delta\left(f(\varphi)\right) \to
\frac{\delta(\varphi-\varphi_0)}{2k q_1 |\sin \varphi_0|}
$$
with $\varphi_0 = \arccos \left(-q_1 / 2k\right)$, which is valid for $q_1 \le 2k$. We can now evaluate the integral over $\varphi$, obtaining
$$
\frac{1}{2kq_1} \int_0^{2\pi} d\theta \, g(\theta,\varphi_0)
$$
If $q_1 > 2k$, the integral evaluates to $0$. 
Evaluating the integral over $\theta$
Work in progress...
