Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 29-32, show that each equation is an identity. \(\tan \left(2 \tan ^{-1} x\right)=\frac{2 x}{1-x^{2}}\)

Step-by-Step Solution

Answer

The equation is an identity.

Step 1: Understanding the Problem

We need to verify if the given equation \( \tan(2 \tan^{-1}(x)) = \frac{2x}{1-x^2} \) is an identity. This means we have to prove it holds true for all values of \( x \) for which both sides are defined.

Step 2: Recognize the Structure

Notice that \( \tan(2 \tan^{-1}(x)) \) involves the tangent double angle formula, which states that \( \tan(2\theta) = \frac{2\tan(\theta)}{1-\tan^2(\theta)} \). Here, \( \theta = \tan^{-1}(x) \).

Step 3: Substitute and Simplify

Substitute \( \theta = \tan^{-1}(x) \) so \( \tan(\theta) = x \). Using the double angle formula, we have \( \tan(2\theta) = \frac{2x}{1-x^2} \). Verify this is exactly the right-hand side of the given equation.

Step 4: Conclusion

Since both expressions \( \tan(2 \tan^{-1}(x)) \) using the double angle formula and \( \frac{2x}{1-x^2} \) simplify to the same form, the given equation is indeed an identity.

Key Concepts

Tangent Double Angle Formula

The tangent double angle formula is a key concept in trigonometry. It helps us find the tangent of double angles. The formula is expressed as:

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

This formula derives from sine and cosine double angle formulas. It converts a complex angle into a simple fraction. The formula is vital when dealing with expressions involving \(\tan(2\theta)\).

To use this formula, substitute \(\theta\) with the angle you are interested in. For example, if \(\theta = \tan^{-1}(x)\), then \(\tan(\theta) = x\). Substitute \(x\) in the formula to find the tangent of double that angle. This process simplifies expressions and solves trigonometric equations.

Inverse Trigonometric Functions

Inverse trigonometric functions are crucial for finding angles when given trigonometric ratios. In our exercise, the focus is on \(\tan^{-1}\), also known as the arctan function. This inverse function determines the angle \(\theta\) whose tangent is a specific value.

• If \( x = \tan(\theta) \), then \( \theta = \tan^{-1}(x) \)

Inverse functions have specific ranges. For \(\tan^{-1}(x)\), the range is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). This range ensures each value of \(x\) corresponds to a single \(\theta\).

Using inverse trigonometric functions allows us to simplify identities and solve equations involving angles. Understanding these concepts helps in verifying identities, as in the given exercise. It showed \(\tan(2 \tan^{-1}(x))\) aligned with the double angle formula.

Algebraic Manipulation

Algebraic manipulation involves transforming expressions into different forms using algebraic rules. This skill is crucial in mathematics to simplify and solve equations. In our exercise, algebraic manipulation helps verify the identity by transforming expressions logically step-by-step.

• Simplification: Reduce expressions by combining like terms or using identities.
• Substitution: Replace variables or expressions consistently to simplify computations.

In the given exercise, we substituted \(\theta\) with \(\tan^{-1}(x)\) to simplify \(\tan(2\theta)\) using the double angle formula. Through algebraic manipulation, we showed both the left and right sides of the equation are identical. Mastering these skills supports solving various mathematical problems efficiently.