Problem 54 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Verify each identity. $$ \begin{aligned} &\tan \left(\frac{\pi}{4}+\alpha\right)-\tan \left(\frac{\pi}{4}-\alpha\right)=2 \tan 2 \alpha\\\ &\text {Hint: Use the result in Exercise } 53 . \end{aligned} $$

Step-by-Step Solution

Answer

The given trigonometric identity has been verified using the tangent sum and difference identities. The process involves setting up the identities for the tangent of the sum and difference of two angles, substituting the values of the angles from the given identity in the verification and simplifying the expression and finally equating the resulting expression to the right-hand side of the identity.

Step 1: Identify and Set up the Identities

First, identify the tangent sum and difference identities. The tangent of sum of two angles \(α\) and \(β\) is given by: \(\tan(α + β) = \frac{\tanα + \tanβ}{1 – \tanα * \tanβ}\). Similarly, the tangent of difference of two angles \(α\) and \(β\) is given by: \(\tan(α - β) = \frac{\tanα - \tanβ}{1 + \tanα * \tanβ}\). We will use these identities by substituting \(α = \frac{\pi}{4}\) and \(β = \alpha\) respectively.

Step 2: Substitute in the Identity

Substitute \(α = \frac{\pi}{4}\) and \(β = α\) in the identities derived in the Step 1. Thus the left hand side of the identity to be proved, which is \(\tan (\frac{\pi}{4} + α) - \tan (\frac{\pi}{4} - α)\), can be written as \(\frac{10α + 1}{1 - \tanα} - \frac{\tanα - 1}{1 + \tanα}\).

Step 3: Simplify the Expression

Simplify the subtraction of two fractions into one fraction, using the rule \(\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}\). On applying this rule, the expression from Step 2 simplifies to \(\frac{2 * \tan\u^2(α)}{1 - \tan\u^2(α)}\).

Step 4: Equating to the Right Hand Side

Now, equate this expression to the right side of the equation given in the problem, which is \(2 * \tan(2α)\). From exercise 53 (given in the Hint) or directly from the formula \(\tan(2α) = \frac{2 * \tanα}{1 - \tan\u^2(α)}\), we find that this expression equals to \(2 * \tan(2α)\). Thus, the identity is proved.

Key Concepts

Tangent Sum and Difference Identities

The tangent sum and difference identities are essential tools in trigonometry for simplifying and computing the tangent of the sum or difference of two angles. These identities allow us to express such tangents in terms of individual angle tangents. Here's how they work:

• The sum identity for tangent is: \(\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha \cdot \tan\beta}\)
• The difference identity for tangent is:\(\tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha \cdot \tan\beta}\)

By substituting specific angle values, such as \(\alpha = \frac{\pi}{4}\) and \(\beta = \alpha\), into these identities, one can effortlessly derive more complex trigonometric identities. These are invaluable when verifying trigonometric equations, as they break down elaborate expressions into simpler base functions.

Angle Addition Formulas

Angle addition formulas generalize the trigonometric identities for sine, cosine, and tangent when dealing with the sum and difference of angles. These formulas are foundational in trigonometry because they reveal relationships between the trigonometric functions of composite angles and their individual angle components. For tangent, there's a specific addition formula:

• The tangent addition formula: \(\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha \cdot \tan\beta}\)
• Similarly, the tangent subtraction formula is: \(\tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha \cdot \tan\beta}\)

These formulas were directly utilized in solving the exercise, allowing the problem to be simplified step-by-step. By substituting values and simplifying using these formulas, complex expressions can fall into familiar forms, making complex trigonometric problems easier to solve.

Trigonometric Simplification

Trigonometric simplification involves breaking down complex trigonometric expressions into simpler forms, making it easier to analyze, prove, or solve them. This process typically employs identities like the sum and difference identities. In the exercise, simplification was achieved as follows:

• First, applying the tangent identities reduced the initial complex expression into a manageable form.
• Then, using the rule for subtracting fractions \(\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}\) helped consolidate multiple fraction expressions into a single equation.
• Lastly, recognizing the form of \(2 \cdot \tan(2\alpha)\) from the simplified equation confirmed that the identity was accurate.

Simplifying trigonometric expressions is a crucial skill, not just for solving equations, but also for understanding and verifying mathematical statements efficiently.