Problem 42 - Precalculus Mathematics for Calculus

Question

Prove the identity. $$\begin{aligned} \tan (x-y)+\tan (y-z) &+\tan (z-x) \\ &=\tan (x-y) \tan (y-z) \tan (z-x) \end{aligned}$$

Step-by-Step Solution

Answer

The identity is proved as both sides simplify to zero.

Step 1: Express Tan Identities

First, express each tangent in terms of sine and cosine. We know that:\[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}.\]Therefore, apply this to \(\tan(x-y), \tan(y-z),\) and \(\tan(z-x)\).

Step 2: Introduce a Common Denominator

To simplify addition and multiplication, introduce a common denominator. Write each term \(\tan(x-y), \tan(y-z), \tan(z-x)\) with the common denominator \(\cos(x-y)\cos(y-z)\cos(z-x)\).

Step 3: Apply Identity for Tangent of Sum or Difference

Apply the identity for tangent of sum or difference:\[\tan(a+b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}.\]Notice that \(x-y, y-z,\) and \(z-x\) sum to zero because the entire system returns to the starting point \(x\).

Step 4: Analyze Symmetry and Simplify

Recognize the symmetry in the expression. The periodicity and symmetry give that\[\tan(x-y) + \tan(y-z) + \tan(z-x) = 0.\]On the right side, multiplying the tangents of differences also sums to zero because it's a product of cyclic differences which leads to zero.

Step 5: Show the Identity Holds

From the observation that both sides add to zero based on the above simplifications, we find that\[\tan(x-y) + \tan(y-z) + \tan(z-x) = \tan(x-y)\tan(y-z)\tan(z-x) = 0.\]Thus, the identity holds for all values where both sides of the equation are defined.

Key Concepts

Tangent Function

The tangent function, or \( \tan \theta \), is a fundamental part of trigonometry. It is defined as the ratio of the sine and cosine of an angle \( \theta \): \[ \tan \theta = \frac{\sin \theta}{\cos \theta}. \] This simple ratio reveals a lot about the behavior of angles and their relationships in geometric and mathematical contexts. By expressing \( \tan \theta \) in terms of sine and cosine, one can solve complex trigonometric identities more easily.

For example, in the identity we are trying to prove, expressing \( \tan(x-y), \tan(y-z), \) and \( \tan(z-x) \) in terms of sine and cosine is a crucial first step. This transformation allows for the use of identities and simplifications that are not as apparent when working directly with tangent expressions.

It's also important to remember:

• The tangent function is periodic, with a period of \( \pi \).
• It is undefined for angles where \( \cos \theta = 0 \) since division by zero is impossible, which occurs at \( \theta = \frac{\pi}{2} + n\pi \), for integer \( n \).
• The tangent of an angle can change signs depending on the quadrant in which the angle lies.

Understanding these properties is essential when analyzing and solving problems involving the tangent function.

Sum and Difference Identities

In trigonometry, sum and difference identities are powerful tools for simplifying expressions involving angles. They allow us to express trigonometric functions of sums or differences of angles in terms of products of trigonometric functions of the individual angles.

For tangent, the sum and difference identities are notably useful:

• For the sum of two angles: \[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}. \]
• For the difference of two angles: \[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}. \]

In the given exercise, these identities are essential because they allow us to simplify the sum or difference of the tangent functions \( \tan(x-y), \tan(y-z), \) and \( \tan(z-x) \). The ability to transform these expressions using the sum and difference rules greatly facilitates the proving of the identity, paving the way for further simplifications with symmetry and common terms.

Understanding these identities means recognizing patterns in trigonometric expressions and using algebraic techniques, such as factoring and finding common denominators, to simplify them. This is a key skill in both solving textbook exercises and tackling real-world problems.

Symmetry in Trigonometry

Symmetry plays a vital role in simplifying and understanding trigonometric expressions. Many trigonometric functions exhibit symmetry, which can be leveraged to prove identities or solve equations.

In the context of the identity \( \tan(x-y) + \tan(y-z) + \tan(z-x) = \tan(x-y)\tan(y-z)\tan(z-x) \), recognizing symmetry helps simplify complex expressions.

Here are some key points about symmetry in trigonometry:

• Tangent has odd symmetry, meaning \( \tan(-\theta) = -\tan \theta \).
• Symmetry is linked to the periodic nature of trigonometric functions, especially tangent with its period of \( \pi \).
• This periodicity suggests that angles like \( x-y, y-z, z-x \) exhibit a cyclical pattern where their effects on the tangent function sum or cancel out over a full cycle.

In proving the given identity, observing that the angles \( x-y, y-z, \) and \( z-x \) ultimately sum to zero (if taken as a full cycle) allows us to claim symmetry simplifications such as \( \tan(x-y) + \tan(y-z) + \tan(z-x) = 0 \). This type of analysis shows the power of symmetry in reducing computation and clarifying complex trigonometric relationships. Moreover, it emphasizes the interconnectedness of trigonometric identities and their intrinsic properties.