Question #40422

This question tests the rigorous application of De Moivre's Theorem and Vieta's formulas while navigating domain restrictions in trigonometric transformations.

The 'Lethal Collision' occurs when students blindly apply Vieta's formulas to the 7th-degree polynomial in $\cos\theta$ without excluding the root $\cos\theta = 0$ (which corresponds to $\theta = \pi/2$), leading to a division-by-zero error when calculating secant and cosecant sums.

Furthermore, a secondary trap awaits those who successfully derive the cubic polynomial in $\tan^2\theta$: they often forget that the mapping from $\theta \in (0, \pi)$ to $\tan^2\theta$ is two-to-one (excluding $\pi/2$), meaning the sum of the roots of the cubic must be multiplied by 2 to account for all valid angles in the specified interval.