Problem 67 - Algebra and Trigonometry

Question

59–76 Prove the identity. $$\tan 3 x=\frac{3 \tan x-\tan ^{3} x}{1-3 \tan ^{2} x}$$

Step-by-Step Solution

Answer

The identity \( \tan 3x = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x} \) holds true.

Step 1: Recall Tangent Triple Angle Formula

The formula for the tangent of a triple angle is given by:\[ \tan(3x) = \frac{3\tan(x) - \tan^3(x)}{1 - 3\tan^2(x)} \]This is the identity we want to prove.

Step 2: Use Tangent Addition Formula

First, use the tangent addition formula: \[ \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \]To find \( \tan(3x) \), consider it as \( \tan(x + 2x) \).

Step 3: Apply Tangent Formula for \( \tan(x + 2x) \)

Apply the tangent addition formula:\[ \tan(x + 2x) = \frac{\tan(x) + \tan(2x)}{1 - \tan(x)\tan(2x)} \]

Step 4: Calculate \( \tan(2x) \)

For \( \tan(2x) \), use the double angle formula: \[ \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} \]

Step 5: Substitute \( \tan(2x) \) in \( \tan(x + 2x) \)

Now substitute \( \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} \) into the equation:\[ \tan(x + 2x) = \frac{\tan(x) + \frac{2\tan(x)}{1 - \tan^2(x)}}{1 - \tan(x) \cdot \frac{2\tan(x)}{1 - \tan^2(x)}} \]

Step 6: Combine Fractions

Combine fractions in both numerator and denominator:\[\tan(x + 2x) = \frac{\frac{\tan(x)(1 - \tan^2(x)) + 2\tan(x)}{1 - \tan^2(x)}}{\frac{1 - 2\tan^2(x)}{1 - \tan^2(x)}}\]

Step 7: Simplify Expression

After clearing fractions, the expression becomes:\[ \tan(3x) = \frac{3\tan(x) - \tan^3(x)}{1 - 3\tan^2(x)} \]This is the required identity and matches the problem statement.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for any angle. These identities help us simplify and solve equations involving trigonometric functions. Understanding these identities is essential for proving other complex trigonometric formulas, like the tangent triple angle formula. A few basic identities include:

• Pythagorean Identity: \( an^2(x) + 1 = ext{sec}^2(x)\)
• Quotient Identity: \( an(x) = \frac{ ext{sin}(x)}{ ext{cos}(x)}\)

These identities are the building blocks for more advanced operations in trigonometry, allowing us to manipulate expressions to easier forms.

Tangent Addition Formula

The tangent addition formula allows us to find the tangent of the sum of two angles. It is given by:

• \(\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}\)

In the context of the exercise, we use this formula to express \(\tan(3x)\) as \(\tan(x + 2x)\). This step breaks down the problem into parts that can be managed more easily using simpler identities. The idea is to express complex terms as sums or differences of smaller terms, making them easier to handle. By using the tangent addition formula, it becomes possible to substitute known values and simplify the overall expression.

Double Angle Formulas

Double angle formulas are used to express trigonometric functions of \(2x\) in terms of \(x\). For the tangent function, the double angle formula is expressed as:

• \(\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}\)

This formula is particularly useful in breaking down the expression for \(\tan(3x)\) into parts that can be more easily managed. By substituting the double angle formula into the expressions used in the tangent addition formula, the problem becomes simpler and straightforward to solve. Understanding and applying these formulas help simplify complicated trigonometric expressions by using relationships between angles and their multiples.