Sum, Difference, and Multiple-Angle Formulas · Superchallenge problem

The product $\left(1 + \sec 20^\circ\right)\left(1 + \sec 40^\circ\right)\left(1 + \sec 80^\circ\right)$ equals $\tan^2\theta$ for exactly one angle $\theta$ with $0^\circ < \theta < 90^\circ$. Find $\theta$, in degrees.