What is the simplified value of the trigonometric expression [(tan 5θ + tan 3θ) / (4 cos 4θ (tan 5θ − tan 3θ))]?

Introduction / Context:
This question involves a nontrivial trigonometric expression containing tan 5θ, tan 3θ and cos 4θ. The goal is to simplify it to a basic trigonometric function. It tests understanding of tangent sum and difference identities and how they relate to sine and cosine expressions.

Given Data / Assumptions:
- Expression: (tan 5θ + tan 3θ) / (4 cos 4θ (tan 5θ − tan 3θ)).
- θ is an angle for which all functions tan 5θ, tan 3θ and cos 4θ are defined and nonzero as required.

Concept / Approach:
We use the identities for tangent of sum and difference expressed in terms of sine and cosine. Specifically, we rewrite tan 5θ and tan 3θ as sin / cos and then form the ratio (tan 5θ + tan 3θ) / (tan 5θ − tan 3θ). This ratio can be converted into a ratio of sines, which simplifies using double angle and multiple angle formulas. Finally, we simplify with the extra factor 4 cos 4θ in the denominator.

Step-by-Step Solution:
Step 1: Write tan 5θ = sin 5θ / cos 5θ and tan 3θ = sin 3θ / cos 3θ.Step 2: Compute tan 5θ + tan 3θ = (sin 5θ cos 3θ + cos 5θ sin 3θ) / (cos 5θ cos 3θ) = sin(5θ + 3θ) / (cos 5θ cos 3θ) = sin 8θ / (cos 5θ cos 3θ).Step 3: Similarly, tan 5θ − tan 3θ = (sin 5θ cos 3θ − cos 5θ sin 3θ) / (cos 5θ cos 3θ) = sin(5θ − 3θ) / (cos 5θ cos 3θ) = sin 2θ / (cos 5θ cos 3θ).Step 4: Therefore, (tan 5θ + tan 3θ) / (tan 5θ − tan 3θ) = sin 8θ / sin 2θ.Step 5: Now the whole expression becomes [sin 8θ / sin 2θ] / [4 cos 4θ] = sin 8θ / (4 sin 2θ cos 4θ).Step 6: Use the identity sin 8θ = 2 sin 4θ cos 4θ and sin 4θ = 2 sin 2θ cos 2θ, so sin 8θ = 4 sin 2θ cos 2θ cos 4θ.Step 7: Substitute to get sin 8θ / (4 sin 2θ cos 4θ) = [4 sin 2θ cos 2θ cos 4θ] / [4 sin 2θ cos 4θ] = cos 2θ.

Verification / Alternative check:
Pick a simple angle, for example θ = 10°, and evaluate the original expression numerically with a calculator. Then compute cos 2θ at the same θ. Both values will match closely, confirming that the simplification is correct.

Why Other Options Are Wrong:
The expression does not simplify to sin 2θ, tan 4θ or cot 2θ. These alternatives usually arise from incomplete simplification, skipping the step that uses sin 8θ and the factor 4 cos 4θ, or mixing up formulas for tan of sum and difference.

Common Pitfalls:
Students sometimes try to apply tan(A + B) formulas directly in a complicated way instead of first converting to sine and cosine. It is also easy to lose track of the factor 4 cos 4θ in the denominator, which is essential to canceling terms and arriving at cos 2θ. Careful step-by-step rewriting avoids these issues.

Final Answer:
The simplified value of the given expression is cos 2θ.