Problem 45 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 37-46: Verify the identity. $$ \frac{1}{\cot \alpha-\cot \beta}=\frac{\sin \alpha \sin \beta}{\sin (\beta-\alpha)} $$

Step-by-Step Solution

Answer

The identity is verified.

Step 1: Rewrite Cotangent Identity

Start by expressing the cotangent in terms of sine and cosine. Recall the identity \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Thus, \( \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \) and \( \cot \beta = \frac{\cos \beta}{\sin \beta} \). Substitute these into the left side of the identity.

Step 2: Simplify the Difference of Two Cotangents

Subtract these cotangents: \( \cot \alpha - \cot \beta = \frac{\cos \alpha}{\sin \alpha} - \frac{\cos \beta}{\sin \beta} \). Use a common denominator to combine the fractions: \( \frac{\cos \alpha \sin \beta - \cos \beta \sin \alpha}{\sin \alpha \sin \beta} \).

Step 3: Recognize the Sine of a Difference Formula

Notice that the numerator \( \cos \alpha \sin \beta - \cos \beta \sin \alpha \) is the equivalent of \( \sin(\beta - \alpha) \) using the sine of a difference formula, \( \sin(x-y) = \sin x \cos y - \cos x \sin y \).

Step 4: Substitute and Simplify

Substitute \( \sin(\beta - \alpha) \) from Step 3 into the expression from Step 2: \( \frac{\sin(\beta - \alpha)}{\sin \alpha \sin \beta} \). The left hand side \( \frac{1}{\cot \alpha - \cot \beta} \) becomes \( \frac{1}{\frac{\sin(\beta - \alpha)}{\sin \alpha \sin \beta}} \), which simplifies to \( \frac{\sin \alpha \sin \beta}{\sin(\beta - \alpha)} \).

Step 5: Verify the Identity

We have shown that both sides of the original identity equal \( \frac{\sin \alpha \sin \beta}{\sin(\beta - \alpha)} \). Hence, the given identity \( \frac{1}{\cot \alpha - \cot \beta} = \frac{\sin \alpha \sin \beta}{\sin(\beta - \alpha)} \) is verified.

Key Concepts

Cotangent Identity

The cotangent identity is an essential trigonometric identity that expresses the cotangent of an angle \( \theta \) in terms of sine and cosine. Specifically, it states \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). This means that cotangent is the ratio of the cosine to the sine of an angle.

Using this identity helps in simplifying expressions involving cotangent by converting them into terms of sine and cosine. For example, if you have \( \cot \alpha - \cot \beta \), you can rewrite it as \( \frac{\cos \alpha}{\sin \alpha} - \frac{\cos \beta}{\sin \beta} \). This transformation is particularly useful when verifying trigonometric identities since many manipulations and simplifications are more convenient in terms of sine and cosine.

In the verification of trigonometric identities, substituting cotangent with its equivalent in sine and cosine (using this identity) often reveals familiar patterns or formulas, aiding in further simplification. This foundational step is key in many trigonometry problems.

Sine of a Difference Formula

The sine of a difference formula is a crucial tool in trigonometry used to expand \( \sin(\beta - \alpha) \). The formula is \( \sin(x-y) = \sin x \cos y - \cos x \sin y \). This formula allows you to express the sine of the difference between two angles in terms of the sines and cosines of the angles themselves.

In the given trigonometric identity problem, recognizing that the expression \( \cos \alpha \sin \beta - \cos \beta \sin \alpha \) can be rewritten using this formula as \( \sin(\beta - \alpha) \) is a pivotal step. This identification simplifies the expression significantly, demonstrating the importance of this formula in restructuring and simplifying trigonometric expressions.

Memorizing and understanding the sine of a difference formula lets you work through complex trigonometric identities efficiently. It helps in converting and combining separate trigonometric terms into a single, simpler expression. This skill is essential not only for solving identities but also for solving equations and proving theorems in trigonometry.

Simplifying Rational Expressions

Simplifying rational expressions is a fundamental algebraic skill, important for both algebra and trigonometry. A rational expression is essentially a fraction where the numerator and or the denominator contain variables. Simplification involves rewriting these expressions in their simplest form.

In trigonometric problems, simplifying expressions involves changing complex fractions into simpler forms that can be easily compared or manipulated. Consider the expression \( \frac{1}{\cot \alpha - \cot \beta} \). This can be transformed into simpler terms, using the identities and definitions of sine and cosine as the tools.

By finding a common denominator—\( \sin \alpha \sin \beta \) in this case—you combine the expressions, which allows for further simplification. This skill is useful in trigonometric problems as it involves recognizing opportunities to apply fundamental identities and perform algebraic operations effectively to reach a result that confirms or supports mathematical identities or equations.