Evaluating the Product of Tangents from 1° to 89°: A Comprehensive Guide

Understanding the value of the product P tan 1° × tan 2° × tan 3° × … × tan 89° involves the use of trigonometric properties and identities. This article provides a step-by-step explanation of how to evaluate this product, making use of the complementary angle property of the tangent function.

Key Property of the Tangent Function

The tangent function has the following property:

tan(90° - x) cot x

Using this property, we can pair the tangents such that each pair simplifies to 1. This approach significantly simplifies the evaluation of the entire product.

Breaking Down the Problem

Consider the product:

P tan 1° × tan 2° × tan 3° × … × tan 89°

We can pair the terms in the product as follows:

tan 1° pairs with tan 89° tan 2° pairs with tan 88° ... tan 44° pairs with tan 46° tan 45° remains unpaired

Applying the Property

For each pair (tan k°, tan 90° - k°), the product simplifies due to the following:

tan k° × tan (90° - k°) tan k° × cot k° 1

This simplification is valid for each pair from (tan 1°, tan 89°), (tan 2°, tan 88°), ..., (tan 44°, tan 46°).

Final Calculation

Combining the simplifications, the product becomes:

P (tan 1° × tan 89°) × (tan 2° × tan 88°) × ... × (tan 44° × tan 46°) × tan 45°

Since each pair simplifies to 1, the product further simplifies to:

P 1 × 1 × ... × 1 × tan 45°

Note that tan 45° 1. Therefore:

P 1 × 1 × ... × 1 × 1 1

Conclusion

The value of the product of tangents from 1° to 89° is:

P 1

This result is derived by leveraging the complementary angle property and the fact that tan 45° 1.

Additional Insights

The problem of evaluating such products often arises in trigonometry and calculus. Understanding these properties can be crucial for solving more complex trigonometric identities and integrals.

By reinforcing and applying the complementary angle property of the tangent function, we can systematically simplify and evaluate products involving tangents of angles from 1° to 89°. This method of breaking down the problem and using fundamental trigonometric identities can be a powerful tool in solving similar problems.