Finding the equilibrium angle for a dipole when another dipole in the vicinity is held fixed

Panofsky and Phillips’ Exercise 11 of Chapter 1:

Consider two coplanar electric dipoles with there centers a fixed distance apart. Show that if the angles the dipoles make with the line joining their centers are $theta$ and $theta’$, and if $theta$ is held fixed,
$$
tantheta = -frac{1}{2}tan theta’
$$
for equilibrium.

What I did:

In the text was proven the potential energy for a pair of dipoles with separation $mathbf r$ between them:
$$
U = frac{1}{4piepsilon_0}left[
frac{mathbf{p_1cdot p_2}}{r^3} – frac{3}{r^5}(mathbf{p_1cdot r})(mathbf{p_2cdot r})
right]
$$
This reduces to $2costhetacostheta’ – sinthetasintheta’$ times some constant. Setting $U'(theta’)=0$ now yields
$$
tantheta=-2tantheta’.
$$

Question: This is obviously wrong (which is easily seen if one tries to visualize the field lines of a dipole). So what has gone awry?