Solving Trigonometric Identities: The Value of tan(π/16)tan(3π/16)tan(5π/16)tan(5π/16)tan(7π/16)

Welcome to this guide on solving a complex trigonometric product identity using the properties of complementary angles and the tangent function. Specifically, we will find the value of the expression (text{tan}left(frac{pi}{16}right) text{tan}left(frac{3pi}{16}right) text{tan}left(frac{5pi}{16}right) text{tan}left(frac{5pi}{16}right) text{tan}left(frac{7pi}{16}right)):

Understanding the Complementary Angle Identity

To solve the given expression, we will use the property of complementary angles in trigonometry, which states that the tangent of an angle is the cotangent of its complementary angle. This identity is fundamental in simplifying trigonometric expressions and will be our primary tool.

Applying the Complementary Angle Identity

The first step in solving the expression is to apply the complementary angle identity for each term:

(text{tan}left(frac{7pi}{16}right) cotleft(frac{pi}{16}right))

(text{tan}left(frac{5pi}{16}right) cotleft(frac{3pi}{16}right))

Substituting these identities into the original expression, we get:

[text{tan}left(frac{pi}{16}right) text{tan}left(frac{3pi}{16}right) text{tan}left(frac{5pi}{16}right) text{tan}left(frac{5pi}{16}right) text{tan}left(frac{7pi}{16}right) text{tan}left(frac{pi}{16}right) cotleft(frac{pi}{16}right) text{tan}left(frac{3pi}{16}right) cotleft(frac{3pi}{16}right)]

Simplifying Using Tan and Cot Identities

Next, we apply the identity that states (text{tan}x cot x 1) for any angle x. This identity will allow us to simplify the expression as follows:

Step 1: Simplify the Product

Using the identity (text{tan}left(frac{pi}{16}right) cotleft(frac{pi}{16}right) 1) and (text{tan}left(frac{3pi}{16}right) cotleft(frac{3pi}{16}right) 1), we can simplify the product:

[1 cdot 1 1]

Step 2: Final Simplification

Therefore, the overall product simplifies to:

[1 cdot 1 1]

Hence, the value of the given expression is 1.

Conclusion

In conclusion, the value of (text{tan}left(frac{pi}{16}right) text{tan}left(frac{3pi}{16}right) text{tan}left(frac{5pi}{16}right) text{tan}left(frac{5pi}{16}right) text{tan}left(frac{7pi}{16}right)) is 1. This result is a direct application of the complementary angle identities and the key properties of the tangent and cotangent functions.

Related Topics and Keywords

Trigonometric Identities - The calculus and algebraic relationships between trigonometric functions that are true for all defined angles. Tangent Product - Expressions involving the product of tangent values of various angles. Complementary Angles - Angles whose sum is 90 degrees (π/2 radians) and whose trigonometric functions are related by identity.

By understanding and applying these concepts, you can solve a wide range of trigonometric problems and deepen your knowledge of advanced mathematics.