Problem 81 - Precalculus Mathematics for Calculus

Question

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 \( \tan 3x = \frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x} \) is proved using angle addition and simplification.

Step 1: Understand the Identity

The identity to prove is \( \tan 3x = \frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x} \). We will use the angle addition formulas to expand and simplify both sides to confirm their equivalence.

Step 2: Use Triple Angle Formula for Tangent

The triple angle formula for tangent states that \( \tan 3x = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x} \). We need to verify this identity step-by-step by expanding the left-hand side using known formulas.

Step 3: Expand \( \tan 3x \) Using Angle Addition

Use the known tangent angle addition formula: \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \). First, consider \( \tan(2x + x) \), which applies as follows: \[ \tan(2x + x) = \frac{\tan 2x + \tan x}{1 - \tan 2x \tan x} \]

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

Use the double angle formula for tangent: \( \tan 2x = \frac{2\tan x}{1 - \tan^2 x} \). Substitute this back into the equation from Step 3.

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

Replace \( \tan 2x \) in Step 3 with its expression from Step 4: \[ \tan(2x + x) = \frac{\frac{2\tan x}{1 - \tan^2 x} + \tan x}{1 - \frac{2\tan x}{1 - \tan^2 x} \tan x} \]. Simplify the equation by combining terms.

Step 6: Simplify the Numerator

Simplify the numerator of the expanded \( \tan(2x + x) \) equation: \[ \frac{2\tan x + \tan x (1 - \tan^2 x)}{1 - \tan^2 x} = \frac{2\tan x + \tan x - \tan^3 x}{1 - \tan^2 x} \]. This becomes \( \frac{3\tan x - \tan^3 x}{1 - \tan^2 x} \).

Step 7: Simplify the Denominator

Simplify the denominator of \( \tan(2x + x) \): \[ 1 - \frac{2\tan^2 x}{1 - \tan^2 x} \]. Combine into a single fraction: \( \frac{1 - 3\tan^2 x}{1 - \tan^2 x} \).

Step 8: Combine the Simplified Fractions

Divide the simplified numerator by the simplified denominator (Step 6 by Step 7): \[ \frac{3\tan x - \tan^3 x}{1 - \tan^2 x} \times \frac{1 - \tan^2 x}{1 - 3\tan^2 x} \]. This simplifies directly to \( \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x} \), verifying the identity.

Key Concepts

Triple Angle Formulas

Triple angle formulas are essential tools in trigonometry that allow you to express trigonometric functions of three times an angle in terms of those functions of the original angle. They simplify complex trigonometric expressions and can help prove identities, like the one given in the original exercise: \[ \tan 3x = \frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x}. \] These formulas are closely related to the multiple angle formulas, which are particularly useful in solving equations and simplifying expressions. The beauty of the triple angle formula for tangent is that it provides a compact way to express \( \tan 3x \) using basic arithmetic operations on \( \tan x \).

• The numerator \( 3 \tan x - \tan^3 x \) combines linear and cubic terms.
• The denominator \( 1 - 3 \tan^2 x \) subtracts a term related to the square of the tangent.

Understanding these terms can help in various applications, including signal processing and complex number calculations.

Tangent Function

The tangent function, denoted as \( \tan(x) \), is one of the basic trigonometric functions. It's defined as the ratio of the sine and cosine of the same angle, so \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). This function helps measure how steep a line is relative to the x-axis. In the unit circle, the tangent of an angle can also be visualized as the length of the segment that is tangential to the circle at one point and perpendicular to the radius. This segment intersects the extension of the tangent radius. The tangent function is periodic and repeats every \( \pi \) radians (or 180 degrees). It has asymptotes (vertical lines that the curve approaches but never touches) where the function is undefined; these occur where \( \cos(x) \) is zero.

• Sharp turns and zero crossings provide information about angle transformations.
• The steepness increases dramatically between zero crossings, depicted by nearing the asymptotes.

Angle Addition Formulas

Angle addition formulas in trigonometry allow us to express trigonometric functions of sums of angles in terms of functions of the individual angles themselves. For the tangent function, the angle addition formula is given by: \[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}. \] This formula plays a crucial role in simplifying and proving identities involving tangents, like our identity of \( \tan 3x \). By breaking down \( 3x \) as \( 2x + x \), and further using the tangent double angle formula for \( 2x \), we can systematically deconstruct the expression using smaller angle transformations. Each breakdown follows easily, substituting smaller components:

• Express \( \tan(2x + x) \) using \( \tan 2x \) and \( \tan x \).
• Substitute \( \tan 2x \) with known formulas and simplify.

These methods showcase the power of angle addition formulas in unlocking more complex trigonometric identities.