Evaluate the product tan 6° × tan 36° × tan 84° × tan 54° × tan 45° using known trigonometric identities and complementary angle relationships. What is the exact value of this product?

Introduction / Context:
This trigonometric problem involves the product of tangent values at several special angles: 6°, 36°, 84°, 54°, and 45°. Recognising complementary angles and using identities can greatly simplify such products, which are popular in advanced aptitude and competition style questions.

Given Data / Assumptions:

• Angles are in degrees.
• We must compute tan 6° × tan 36° × tan 84° × tan 54° × tan 45°.
• Standard relationships for tangent of complementary angles apply.

Concept / Approach:
The key idea is that tan(90° − θ) = cot θ and that tan 45° = 1. So some terms can be paired as complementary angles. Specifically, 84° and 6° are complementary, as are 54° and 36°. Pairing them allows the product to be written in terms of tan θ × tan(90° − θ) which simplifies neatly using the identity tan θ × cot θ = 1.

Step-by-Step Solution:
Group the terms as (tan 6° × tan 84°) × (tan 36° × tan 54°) × tan 45°. Note that 84° = 90° − 6° and 54° = 90° − 36°. Use tan(90° − θ) = cot θ, so tan 84° = cot 6° and tan 54° = cot 36°. Thus tan 6° × tan 84° = tan 6° × cot 6° = 1. Similarly, tan 36° × tan 54° = tan 36° × cot 36° = 1. We also know tan 45° = 1. So the overall product is 1 × 1 × 1 = 1.
Verification / Alternative check:
If you use a calculator or approximate values, each pair tan 6° × tan 84° and tan 36° × tan 54° will be extremely close to 1, and tan 45° is exactly 1. Numerically multiplying all five values will yield a result very close to 1, confirming the algebraic simplification.

Why Other Options Are Wrong:
Options a, b, and d represent fractions less than 1 and do not match the final simplified product. Option e (2) is larger than 1 and has no basis in the complementary angle identities used here. The structure of the product almost guarantees a clean value of 1 once the complementary relationships are recognised.

Common Pitfalls:
A common mistake is to try to evaluate each tangent in decimal form without noticing the complementary patterns, which can lead to rounding errors and wasted time. Another error is misapplying the identity tan(90° − θ) = cot θ with an incorrect angle. Recognising complementary angles is the crucial insight for this problem.

Final Answer:
The exact value of the product tan 6° tan 36° tan 84° tan 54° tan 45° is 1.