Problem 29 - Precalculus Mathematics for Calculus

Question

Prove the identity. $$\tan (x-\pi)=\tan x$$

Step-by-Step Solution

Answer

The identity \( \tan(x - \pi) = \tan x \) is proven using the tangent subtraction formula.

Step 1: Understanding the Problem

We are asked to prove the trigonometric identity \( \tan(x - \pi) = \tan x \). This involves showing that the tangent of the difference angle \( \pi \) results in the same value as the tangent of \( x \).

Step 2: Using Angle Subtraction Formula

We'll use the formula for tangent of a difference: \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \). Set \( A = x \) and \( B = \pi \).

Step 3: Simplifying \( \tan \pi \)

Since \( \tan \pi = 0 \), substitute into the formula: \( \tan(x - \pi) = \frac{\tan x - 0}{1 + \tan x \cdot 0} \).

Step 4: Simplifying the Expression

The expression becomes \( \frac{\tan x}{1 + 0} = \tan x \). This confirms that \( \tan(x - \pi) = \tan x \).

Step 5: Conclusion

We have used the tangent subtraction formula to show that subtracting an angle of \( \pi \) from \( x \) does not change the value of \( \tan x \), thus proving the identity.

Key Concepts

Angle Subtraction Formula

The angle subtraction formula is a pivotal concept in trigonometry. It helps in finding the trigonometric function values of the difference between two angles. For the tangent function, the angle subtraction formula is given by:

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

This formula is incredibly useful when you need to simplify or manipulate expressions involving angle differences. For instance, in the exercise provided, it allows us to prove the identity \( \tan(x - \pi) = \tan x \) by substituting appropriate values for \( A \) and \( B \). By assigning \( A = x \) and \( B = \pi \), and knowing that \( \tan \pi = 0 \), we effectively simplify this identity.
Using this formula allows students to not only validate trigonometric identities but also understand the cyclic nature of the tangent function and its periodicity with a period of \( \pi \).

Tangent Function

The tangent function, denoted as \( \tan \), is one of the primary trigonometric functions and is defined as the ratio of the sine to the cosine of an angle. In mathematical terms, this is expressed as:

• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

The tangent function has some unique properties among trigonometric functions. Notably, its periodicity differs from that of sine and cosine, as it repeats every \( \pi \) rather than \( 2\pi \). This periodicity means that \( \tan(\theta + \pi) = \tan(\theta) \). This property highlights the essential characteristic used in the problem exercise—demonstrating that subtracting \( \pi \) from an angle does not change the value of the tangent. This understanding can deepen your comprehension of periodicity and help simplify trigonometric identities effectively.
Understanding the tangent function, especially its periodicity and symmetry, is crucial in both solving specific trigonometric problems and appreciating how trigonometry elegantly describes circular motion and wave behavior.

Simplifying Expressions

Simplifying expressions in trigonometry often involves applying identities and formulas to transform a complex expression into a simpler or more manageable form. This process is key in problem-solving.
In our exercise, the expression \( \tan(x - \pi) \) was simplified using the angle subtraction formula. With \( \tan \pi = 0 \), substituting into the formula results in:

• \( \tan(x - \pi) = \frac{\tan x - 0}{1 + \tan x \cdot 0} = \tan x \)

This simplification relies on understanding basic properties of the trigonometric functions and their behavior at specific angles, in this case, knowing that \( \tan \pi = 0 \). It underscores the importance of recognizing when trigonometric identities apply and how they can reduce complexity.
While this might seem straightforward, it's a fundamental skill in trigonometry. Simplifying expressions helps solve equations analytically, make logical deductions, or even verify solutions. Mastering this skill is not only vital in trigonometry but across all areas of mathematics where complex expressions need to be transformed into simpler ones.

Precalculus

Precalculus acts as a bridge between algebra and calculus. It encompasses a variety of topics, including trigonometry, which are crucial for understanding calculus concepts.
In the context of this exercise, precalculus introduces and elaborates on the trigonometric identities and the angle subtraction formula. These are foundational tools necessary for calculus, where rate of change and area under a curve depend on trigonometric integrals and derivatives, like those of sine, cosine, and tangent functions.

• Trigonometric identities simplify complex calculations and provide solutions to problems that would otherwise require more effort.
• Understanding functions like tangent and their periodic behaviors lays the groundwork for studying oscillations and wave functions in calculus.

Grasping these concepts in precalculus is not just about passing the course but preparing for mathematical challenges ahead. It builds analytical skills and a deeper appreciation for the beauty and utility of mathematics.