Define the family of densities:

$$p(\phi;\theta) = \Big(f\big(\hspace{-1pt}\cos(\phi-\theta)\big) - f\big(\hspace{-1pt}\cos(\phi+\theta)\big)\Big)\hspace{0.5pt} \frac{\sin(2\phi)}{\sin(2\theta)}, \quad 0 \le \phi,\theta\le \pi/2$$

where $f(x)=g(x^2)$ with $g$ non-negative, increasing, and convex or concave, on $[0,\infty)$. (Interestingly these densities indeed have the same measure for all $\theta$ whenever $g$ is concave or convex, which can be shown by an integral representation.)

Show that $p(\phi;\theta)$ has a monotone likelihood ratio (decreasing or increasing depending on the concavity or convexity of $g$). I.e., for $0\le\theta_1 < \theta_2\le\pi/2$:

$$ h(\phi) = \frac{f\big(\hspace{-1pt}\cos(\phi-\theta_2)\big) - f\big(\hspace{-1pt}\cos(\phi+\theta_2)\big)}{f\big(\hspace{-1pt}\cos(\phi-\theta_1)\big) - f\big(\hspace{-1pt}\cos(\phi+\theta_1)\big)}$$

is monotonic on $[0,\pi/2]$.