Evaluating the Trigonometric Expression: Tan(2 tan^-1(1/5) - π/4)

Understanding and evaluating trigonometric expressions is a fundamental skill in advanced mathematics. One such expression is tan(2 tan^{-1}(1/5) - frac{pi}{4}). This article will walk you through the process of evaluating this expression step-by-step, using various trigonometric identities and properties.

Step-by-Step Evaluation

Let's break down the expression into smaller, manageable parts and utilize the double angle formula for tangent and the tangent subtraction formula.

Step 1: Introducing the Variable

Let x tan^{-1}(1/5). This implies:

tan x frac{1}{5}

Step 2: Applying the Double Angle Formula

The double angle formula for tangent is:

tan 2x frac{2 tan x}{1 - tan^2 x}

Substituting tan x frac{1}{5}, we get:

tan 2x frac{2 cdot frac{1}{5}}{1 - left( frac{1}{5} right)^2} frac{frac{2}{5}}{1 - frac{1}{25}} frac{frac{2}{5}}{frac{24}{25}} frac{2}{5} cdot frac{25}{24} frac{10}{24} frac{5}{12}

Step 3: Using the Tangent Subtraction Formula

The tangent subtraction formula is:

tan (A - B) frac{tan A - tan B}{1 tan A tan B}

Let A 2x and B frac{pi}{4}. Since tan(frac{pi}{4}) 1, we have:

tan(2x - frac{pi}{4}) frac{tan 2x - 1}{1 tan 2x cdot 1}

Substituting tan 2x frac{5}{12}, we get:

tan(2x - frac{pi}{4}) frac{frac{5}{12} - 1}{1 frac{5}{12}} frac{frac{5}{12} - frac{12}{12}}{frac{12}{12} frac{5}{12}} frac{frac{-7}{12}}{frac{17}{12}} frac{-7}{17}

Thus, the final result is:

boxed{frac{-7}{17}}

Alternative Methods and Simplifications

There are alternative methods to solve this problem without using the double angle formula directly. Here's a simplified version using the trigonometric identity:

tan(A - B) frac{tan A - tan B}{1 tan A tan B}

Applying this identity:

tan(2 tan^{-1}(1/5) - frac{pi}{4}) frac{tan(2 tan^{-1}(1/5)) - 1}{1 tan(2 tan^{-1}(1/5))}

Using the two-argument arctangent formula:

tan(2 tan^{-1}(x)) frac{2x}{1 - x^2}

Substituting x frac{1}{5}, we get:

tan(2 tan^{-1}(1/5)) frac{2 cdot frac{1}{5}}{1 - left( frac{1}{5} right)^2} frac{frac{2}{5}}{1 - frac{1}{25}} frac{frac{2}{5}}{frac{24}{25}} frac{2}{5} cdot frac{25}{24} frac{10}{24} frac{5}{12}

Now, using the tangent subtraction formula:

tan(2 tan^{-1}(1/5) - frac{pi}{4}) frac{frac{5}{12} - 1}{1 frac{5}{12}} frac{frac{5}{12} - frac{12}{12}}{1 frac{5}{12}} frac{frac{-7}{12}}{frac{17}{12}} frac{-7}{17}

Therefore, the final result remains:

boxed{frac{-7}{17}}

Key Trigonometric Identities and Properties

In the evaluation of the given expression, we utilized several key identities and properties:

Double Angle Formula for Tangent: (tan 2x frac{2 tan x}{1 - tan^2 x}) Tangent Subtraction Formula: (tan (A - B) frac{tan A - tan B}{1 tan A tan B}) Two-Argument Arctangent Formula: (tan(2 tan^{-1}(x)) frac{2x}{1 - x^2})

Understanding and applying these identities and properties correctly is crucial to solving such trigonometric expressions.

Practice and Resources

To further enhance your understanding and proficiency, you can practice similar problems, review the relevant sections in your textbook, and explore additional resources such as online tutorials and quizzes. The more you practice, the better you will become in tackling complex trigonometric expressions.