Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

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 because both sides simplify to \( \frac{2x}{1-x^2} \) using trigonometric identities.

Step 1: Understand the Identity to Prove

We need to prove that the expression \( \tan(2 \tan^{-1}(x)) \) is equivalent to \( \frac{2x}{1-x^2} \). This means that for any value of \( x \) where the expressions are defined, both sides should be equal.

Step 2: Recall Double Angle Formula for Tangent

Recall the double angle formula for tangent: \( \tan(2\theta) = \frac{2\tan(\theta)}{1-\tan^2(\theta)} \). This formula will be applied in the context of \( \theta = \tan^{-1}(x) \).

Step 3: Apply Definition of Inverse Tangent

Since \( \theta = \tan^{-1}(x) \), it implies that \( \tan(\theta) = x \). This is by definition of the inverse tangent function.

Step 4: Substitute into Double Angle Formula

Substitute \( x \) for \( \tan(\theta) \) in the double angle formula: \[ \tan(2\tan^{-1}(x)) = \frac{2x}{1-x^2}. \] This substitution verifies that the given equation is indeed an identity.

Step 5: Conclusion of the Proof

Both sides of the original problem simplify to the same expression, \( \frac{2x}{1-x^2} \), using the trigonometric identities for tangent. Therefore, the given equation is an identity.

Key Concepts

Double Angle Formula

The double angle formula is an essential tool in trigonometry that helps us express trigonometric functions of double angles in terms of single angles. For the tangent function, the double angle formula is expressed as:\[ \tan(2\theta) = \frac{2\tan(\theta)}{1-\tan^2(\theta)} \]This formula is particularly useful when analyzing expressions involving trigonometric functions, as it allows for simplification and transformation into more usable forms. In the context of our exercise, we apply the double angle formula to \(\tan(2\tan^{-1}(x))\). By setting \(\theta = \tan^{-1}(x)\), the formula helps us in proving that the given equation is an identity. This is possible because we substitute \(x\) for \(\tan(\theta)\), transforming the equation as follows:

• Replacing \(\tan(\theta)\) with \(x\), we get \(\tan(2\tan^{-1}(x)) = \frac{2x}{1-x^2}\).

Understanding and effectively using the double angle formula can significantly simplify complex trigonometric expressions, allowing for clearer insights and solutions.

Inverse Trigonometric Functions

Inverse trigonometric functions are the reverse functions of the standard trigonometric functions. For example, the inverse tangent function, denoted as \(\tan^{-1}(x)\) or \(\text{arctan}(x)\), gives the angle \(\theta\) such that \(\tan(\theta) = x\). These functions are crucial in trigonometry as they allow us to find angles when given trigonometric values.
In our exercise, \(\theta = \tan^{-1}(x)\) serves a pivotal role. By this definition, we know that \(\tan(\theta) = x\). Using this relationship, we can follow through with the double angle formula for tangent.
Inverse trigonometric functions not only aid in proving identities, but they also help in solving equations and analyzing geometric figures. They extend the capabilities of regular trigonometric calculations by converting values back to angles, providing deeper insight into problems featuring trigonometric expressions.

Proof Techniques

Proof techniques involve various strategies used to demonstrate the truth of mathematical statements. In trigonometry, proving identities is a common task, and one effective method is strategic substitution coupled with known formulas and definitions.
Our exercise showcases this approach well. The goal is to transform and simplify one or both sides of an equation until they appear identical. Here's how we approached the proof in this instance:

• We understood the equation to be proven as an identity: \(\tan(2\tan^{-1}(x)) = \frac{2x}{1-x^2}\).
• We used the known double angle formula for tangent to guide our transformations.
• By substituting \(x\) for \(\tan(\theta)\) where \(\theta = \tan^{-1}(x)\), the left side of the equation matched the right side.

This clear, step-by-step substitution and evaluation is a fundamental proof technique. It utilizes known identities and logical deductions to achieve a simplified conclusion. Practicing such methods enhances problem-solving skills and aids in mastering trigonometric concepts.