Problem 35 - Algebra and Trigonometry

Question

Prove the identity. $$ \cot (x-y)=\frac{\cot x \cot y+1}{\cot y-\cot x} $$

Step-by-Step Solution

Answer

The identity is proven by expressing \( \cot(x-y) \) using tangent difference and simplifying.

Step 1: Recall cotangent and tangent identities

We start by recalling that the cotangent of an angle is the reciprocal of the tangent: \( \cot \theta = \frac{1}{\tan \theta} \). Additionally, the tangent of a difference is \( \tan(x-y) = \frac{\tan x - \tan y}{1 + \tan x \tan y} \).

Step 2: Express cotangent of difference in terms of tangent

Using the definition of cotangent, we express \( \cot(x-y) \):\[ \cot(x-y) = \frac{1}{\tan(x-y)} = \frac{1}{\frac{\tan x - \tan y}{1 + \tan x \tan y}} = \frac{1 + \tan x \tan y}{\tan x - \tan y}. \]

Step 3: Rewrite using cotangent of x and y

Notice that \( \tan x = \frac{1}{\cot x} \) and \( \tan y = \frac{1}{\cot y} \). Substitute these into the formula: \[ \frac{1 + \frac{1}{\cot x} \cdot \frac{1}{\cot y}}{\frac{1}{\cot x} - \frac{1}{\cot y}} = \frac{1 + \frac{1}{\cot x \cot y}}{\frac{\cot y - \cot x}{\cot x \cot y}}. \]

Step 4: Simplify the expression

Simplify the expression by multiplying numerator and denominator by \( \cot x \cot y \): \[ \frac{\cot x \cot y + 1}{\cot y - \cot x}. \] This matches the right-hand side of the identity to be proved.

Key Concepts

Cotangent Identities

Cotangent identities are an integral part of trigonometry. The cotangent of an angle, \( \theta \), is defined as the reciprocal of the tangent:\( \cot \theta = \frac{1}{\tan \theta} \). This simple identity allows you to convert between cotangent and tangent as needed.
For example, in problems involving both cotangent and tangent, you can switch back and forth using this relationship. It simplifies problems where you need to manipulate or prove expressions involving \( \cot \theta \).
Understanding this identity is crucial when dealing with complex trigonometric formulas, as it provides a straightforward method to transform expressions for easier manipulation.

Tangent of a Difference

The formula for tangent of a difference is a key concept in solving trigonometric problems that involve the difference of two angles.This identity is expressed as: \( \tan(x-y) = \frac{\tan x - \tan y}{1 + \tan x \tan y} \).
This formula helps in breaking down expressions consisting of differences of angles, making it easier to analyze the trigonometric behavior of those angles.
Without this identity, handling expressions with \( \tan(x-y) \) would be much more challenging, as these differences are very common in calculations related to wave motion, oscillations, and alternating currents.
By understanding and applying this identity, students can manage more complicated expressions by rewriting them in terms of single-angle tangent functions.

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in solving trigonometric identities. It refers to the process of using algebraic techniques to simplify, rearrange, and ultimately prove trigonometric expressions.
In the given exercise, we carried out several algebraic manipulations, such as expressing the cotangent in terms of tangent and simplifying complex fractions.

• Identify initial expressions that can be transformed.
• Substitute known identities, like converting tangent to its reciprocal cotangent.
• Simplify complex fractions by multiplying numerator and denominator appropriately.

Mastering algebraic manipulation allows you to seamlessly switch between trigonometric forms and provides the toolset needed in proofs and problem-solving across various levels of mathematics. It's essential to practice these skills regularly to gain proficiency.

Proofs in Trigonometry

Proofs are a major aspect of trigonometry, allowing for the verification of identities and relationships between angles and functions. In trigonometry, proofs often require a blend of known identities and fresh algebraic manipulation.
Begin with recognizing and recalling relevant trigonometric identities. For the given problem, identities related to tangent and cotangent were essential.
Next, apply these identities systematically to transform the expression step by step. Always aim to express terms in a uniform way, such as using common trigonometric functions.
Proofs are about logical reasoning. It involves connecting each step based on known mathematical principles to show an identity holds universally.

• State what you're trying to prove clearly.
• Recall related identities that might be useful.
• Transform the expression using algebraic and trigonometric relationships.

Remember, the goal of proofs in trigonometry is not just finding a solution, but validating that solution universally. Each proof strengthens understanding of how trigonometric functions relate to each other.