Problem 36 - Precalculus: Functions and Graphs

Question

Exer. \(25-36:\) Verify the reduction formula. $$\tan (x+\pi)=\tan x$$

Step-by-Step Solution

Answer

\(\tan(x + \pi) = \tan x\) is verified by using the tangent addition formula and simplifying.

Step 1: Understand the Problem

The goal is to verify the reduction formula for tangent, which states that \(\tan(x + \pi) = \tan x\). This requires us to demonstrate that both sides of the equation are equal for any angle \(x\).

Step 2: Recall the Tangent Addition Formula

Recall the tangent addition formula: \(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\). We will use this formula to express \(\tan(x + \pi)\).

Step 3: Apply the Tangent Addition Formula

Set \(a = x\) and \(b = \pi\) in the tangent addition formula, so \(\tan(x + \pi) = \frac{\tan x + \tan \pi}{1 - \tan x \tan \pi}\). Since \(\tan \pi = 0\), substitute this into the expression.

Step 4: Simplify the Expression

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

Step 5: Verify the Simplification

Realize that the expression \(\frac{\tan x}{1}\) simplifies directly to \(\tan x\). This confirms that \(\tan(x + \pi) = \tan x\) is true for any angle \(x\).

Step 6: Conclusion

Therefore, the reduction formula \(\tan(x + \pi) = \tan x\) is verified to be correct. We've shown through steps that both sides are indeed equal.

Key Concepts

Tangent Function

The tangent function is one of the fundamental trigonometric functions. It is generally denoted as \( \tan(x) \) and defined as the ratio of the sine and cosine of an angle, given by \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
This function is periodic with a period of \( \pi \), which implies that \( \tan(x + \pi) = \tan(x) \) for any angle \( x \).
If you think of the tangent in terms of a right triangle, it represents the ratio of the length of the opposite side to the adjacent side to the angle in question.
Understanding the periodic nature of tangent helps when working with trigonometric identities and solving equations related to periodic functions.

Angle Addition Formula

The angle addition formula is an essential identity in trigonometry. It allows us to find the tangent of a sum of two angles. The formula is expressed as \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \).
This formula can be very useful when dealing with complex angles, as it breaks down complex expressions into more manageable parts.
In the context of our exercise, we are interested in \( \tan(x + \pi) \). By setting \( a = x \) and \( b = \pi \), the formula helps us evaluate \( \tan(x + \pi) \), which illustrates the periodic nature and further verifies reduction formulas.

• The formula is derived from the sine and cosine angle addition identities.
• It is extensively used in calculus, engineering, and physics.

Reduction Formula

The reduction formula refers to trigonometric identities that simplify angles, usually involving adding or subtracting full rotations (like \( \pi \) or \( 2\pi \)).
Specifically, the formula \( \tan(x + \pi) = \tan(x) \) is a reduction formula.
These formulas are shortcuts which simplify computations and allow us to reduce more complex angle expressions to simpler evaluations.

• They exploit the periodic properties of trigonometric functions.
• Understanding reduction formulas is crucial for simplifying problems and verifying equations like the one in the exercise.

Simplification

Simplification in trigonometry often involves reducing complex expressions using identities.
In the exercise, simplification involves using the tangent addition formula and noting that \( \tan(\pi) = 0 \).
This allows the expression \( \tan(x + \pi) = \frac{\tan x + 0}{1 - \tan x \cdot 0} \) to be simplified to \( \tan(x) \).
Simplifying expressions is a fundamental skill in mathematics, making problems easier to solve by breaking them down into basic components.

• It helps in verifying trigonometric identities by reducing them to known results.
• Common techniques include factoring, cancelling, and applying known trigonometric identities.

Verification

Verification is the process of proving that a mathematical statement, like a formula or equation, holds true.
In the context of our problem, we verified \( \tan(x + \pi) = \tan(x) \) using the tangent addition formula and simplification.
This involves demonstrating that both sides of the equation yield the same results for all valid values of \( x \).
Verification is not only about calculation, but understanding the reasons behind why an identity holds.

• It provides confidence in using these identities across various applications.
• Verification often involves a step-by-step approach, confirming that each transformation or simplification is valid.