Problem 61 - Precalculus

Question

For the following exercises, simplify the given expression. $$ \frac{\tan \left(\frac{3}{2} x\right)-\tan \left(\frac{7}{5} x\right)}{1+\tan \left(\frac{3}{2} x\right) \tan \left(\frac{7}{5} x\right)} $$

Step-by-Step Solution

Answer

The simplified expression is \( \tan \left( \frac{1}{10}x \right) \).

Step 1: Identify the Trigonometric Identity

To simplify the given expression, we recognize that it resembles the tangent difference identity: \[ \frac{\tan A - \tan B}{1 + \tan A \tan B} = \tan(A - B) \]. This implies that the given expression can be simplified using this identity for \( A = \frac{3}{2}x \) and \( B = \frac{7}{5}x \).

Step 2: Apply the Identity

We apply the tangent difference identity to simplify the expression: \[ \tan \left( \frac{3}{2}x - \frac{7}{5}x \right) \]. Now, we proceed to simplify the expression inside the tangent function.

Step 3: Simplify the Angle Argument

Simplify the angle \( \frac{3}{2}x - \frac{7}{5}x \) by finding a common denominator. Convert \( \frac{3}{2} \) and \( \frac{7}{5} \) to have the common denominator 10. Thus, \( \frac{3}{2} = \frac{15}{10} \) and \( \frac{7}{5} = \frac{14}{10} \). Now compute: \[ \frac{15}{10}x - \frac{14}{10}x = \frac{1}{10}x \].

Step 4: Write the Simplified Expression

With the simplified angle \( \frac{1}{10}x \), the entire tangent expression simplifies to \( \tan \left( \frac{1}{10}x \right) \).

Key Concepts

Tangent Difference Identity

Understanding trigonometric identities is crucial to simplifying trigonometric expressions. The tangent difference identity is a useful tool for simplifying expressions involving the difference of two tangent functions. It states:

• \( \frac{\tan A - \tan B}{1 + \tan A \tan B} = \tan(A - B) \)

This identity allows us to transform a more complex expression into a simpler one, which is often easier to handle in calculations. In the original exercise, the expression given resembles the left side of this identity. By recognizing this form, we replaced it with the right side of the equation, \( \tan(A - B) \), making the problem much simpler to solve.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves recognizing patterns and applying identities to reduce complex expressions to simpler forms. In the exercise, once the tangent difference identity was applied, the problem required simplifying the angle \( A - B \).

• This often involves finding a common denominator.
• Common denominators make arithmetic with fractions straightforward.

With \( A = \frac{3}{2}x \) and \( B = \frac{7}{5}x \), we converted these fractions to have a common denominator of 10. This step is crucial because it allows us to directly subtract the two fractions. By simplifying the angle, the trigonometric expression naturally becomes easier to work with.

Angle Simplification

The angle simplification step is crucial in transforming trigonometric expressions. It involves breaking down fractions and performing arithmetic operations to combine them. Here’s how we did it:

• Convert \( \frac{3}{2} \) to \( \frac{15}{10} \) and \( \frac{7}{5} \) to \( \frac{14}{10} \).
• Subtract these to find \( A - B \), leading to \( \frac{1}{10}x \).
• This step directly simplifies the tangent expression to \( \tan(\frac{1}{10}x) \).

This simplified angle is the key to reducing the problem to a more manageable expression. Understanding how to manipulate and combine fractions is fundamental in many areas of mathematics, especially when working with trigonometric identities and expressions.