1. **State the problem:** Evaluate the triple integral $$\int_{\frac{\pi}{2}}^{5} \int_0^0 \int_0^0 r \sin \theta \sec^2 \phi \cos \theta \, dr \, d\theta \, d\phi.$$ 2. **Analyze the limits:** The middle integral is from 0 to 0 over $\theta$, and the innermost integral is from 0 to 0 over $r$. Integrating over an interval of zero length means the integral evaluates to zero immediately, regardless of the integrand. 3. **Conclusion:** Since the integrals over $r$ and $\theta$ are over zero-length intervals, the entire triple integral evaluates to zero. **Final answer:** $$0$$