Problem 59 - Contemporary Precalculus

Question

$$\text { Prove the identity.}$$ $$\tan (x+\pi)=\tan x$$

Step-by-Step Solution

Answer

Question: Prove the trigonometric identity $$\tan(x+\pi) = \tan x$$. Answer: By using the addition formula for tangent, $$\tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}$$, and replacing a with x and b with π, we can simplify the equation and find that $$\tan(x+\pi) = \tan x$$.

Step 1: Write the addition formula for tangent

We will write down the addition formula for tangent: $$\tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}$$

Step 2: Replace a with x and b with π

Now, we will replace a with x and b with π: $$\tan(x+\pi) = \frac{\tan x + \tan \pi}{1 - \tan x \tan \pi}$$

Step 3: Evaluate the tangent of π

We know that the tangent of π is 0: $$\tan \pi = 0$$

Step 4: Substitute the value of \(\tan \pi\) into the equation

Replace the value of \(\tan \pi\) in the equation: $$\tan(x+\pi) = \frac{\tan x + 0}{1 - \tan x \cdot 0}$$

Step 5: Simplify the equation

Simplify the equation by removing the 0 terms: $$\tan(x+\pi) = \frac{\tan x}{1}$$

Step 6: Final result

Finally, we have the desired result: $$\tan(x+\pi) = \tan x$$ This proves the identity \(\tan(x+\pi) = \tan x\).

Key Concepts

Tangent Addition Formula

Understanding the tangent addition formula is crucial in proving trigonometric identities, particularly when dealing with angles that involve the sum or difference of two separate angles. Trigonometry often requires us to evaluate the tangent of an angle that is expressed as a sum, such as \( x + \pi \).

The general form of the tangent addition formula is:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
This particular formula plays a vital role in simplifying complex expressions and in proving various trigonometric identities. It draws from the sine and cosine addition formulas and provides a way to break down the tangent of a sum into more manageable parts.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. One of the key functions in trigonometry is the tangent, which is the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle. The tangent function, along with sine and cosine functions, are the primary trigonometric functions used to describe various properties of angles and triangles.

In applying trigonometry to solve problems, it's important to understand the periodic nature of the trigonometric functions, particularly how they behave over the range of angles from \(0\) to \(2\pi\). This knowledge is essential for proving identities like \(\tan(x + \pi) = \tan x\), where \(\pi\) signifies a half-rotation around the unit circle, which changes the sign of the sine and cosine but leaves the tangent unchanged due to its ratio nature.

Precalculus

Precalculus is an advanced mathematical course that prepares students for calculus. It encompasses a variety of topics, including algebra, geometry, and mathematical analysis, with trigonometry being a significant part of the curriculum. Precalculus lays the foundation for understanding limits, derivatives, and integrals, which are core concepts in calculus.

In precalculus, students first encounter the tangent addition formula and other trigonometric identities. Proving these identities requires algebraic manipulation and a deep understanding of trigonometric functions and their properties. Students must understand the unit circle and how angles in standard position relate to the trigonometric functions. Precalculus develops these critical thinking skills by challenging students with exercises designed to deepen their understanding of mathematical relationships, hence preparing them for the complexities of calculus.