To solve this problem, we need to analyze the expression involving multiple terms with squares and products of tangent functions at specific angles. The expression given is:

$\tan^2\frac{\pi}{16} \cdot \tan^2\frac{3\pi}{16} + \tan^2\frac{\pi}{16} \cdot \tan^2\frac{5\pi}{16} + \tan^2\frac{\pi}{16} \cdot \tan^2\frac{7\pi}{16} + \tan^2\frac{3\pi}{16} \cdot \tan^2\frac{5\pi}{16} + \tan^2\frac{3\pi}{16} \cdot \tan^2\frac{7\pi}{16} + \tan^2\frac{5\pi}{16} \cdot \tan^2\frac{7\pi}{16}$

Approach

This problem can be approached using trigonometric identities and properties of tangent functions at specific angles. Notably, the angles $\frac{\pi}{16}, \frac{3\pi}{16}, \frac{5\pi}{16},$ and $\frac{7\pi}{16}$ are symmetric within the first quadrant, and their properties can simplify the calculations.

Calculation

It turns out that a well-known trigonometric identity simplifies this expression to a constant sum:

$\tan^2\frac{\pi}{16} \cdot \tan^2\frac{3\pi}{16} + \tan^2\frac{\pi}{16} \cdot \tan^2\frac{5\pi}{16} + \tan^2\frac{\pi}{16} \cdot \tan^2\frac{7\pi}{16} + \tan^2\frac{3\pi}{16} \cdot \tan^2\frac{5\pi}{16} + \tan^2\frac{3\pi}{16} \cdot \tan^2\frac{7\pi}{16} + \tan^2\frac{5\pi}{16} \cdot \tan^2\frac{7\pi}{16} = 84$

Thus, the correct answer is:

(E) 84

Would you like a more detailed explanation of how this identity applies?

1. What trigonometric identities could simplify products of tangent functions?
2. How does symmetry in trigonometric functions help solve this?
3. Are there general formulas for similar problems involving products of trigonometric functions?
4. Can this problem be solved without memorizing identities?
5. What are other approaches to calculate values for tangent functions at specific angles?

Tip: Recognizing angle patterns in trigonometric expressions can often simplify complex problems.