Problem 87 - Contemporary Precalculus

Question

Prove the identity. $$\frac{\sin x+\sin y}{\cos x-\cos y}=-\cot \left(\frac{x-y}{2}\right)$$

Step-by-Step Solution

Answer

Based on the solution provided, the short answer is: To prove the trigonometric identity, we first rewrote cotangent in terms of sine and cosine. Next, we applied sum-to-product formulas to both the numerator and denominator of the left side of the equation. Finally, we simplified factors on both sides of the equation until they were equivalent, confirming the identity.

Step 1: Express the identity in terms of sine and cosine functions

Since the cotangent function is defined as \(\cot(θ) = \frac{\cos(θ)}{\sin(θ)}\), we can rewrite the right side of the given identity as: $$-\cot \left(\frac{x-y}{2}\right) = -\frac{\cos(\frac{x-y}{2})}{\sin(\frac{x-y}{2})}$$ So, the identity now becomes: $$\frac{\sin x+\sin y}{\cos x-\cos y} = -\frac{\cos(\frac{x-y}{2})}{\sin(\frac{x-y}{2})}$$

Step 2: Apply sum-to-product formulas

To make further progress, let's use the sum-to-product formulas for sine and cosine: $$\sin x+\sin y = 2\sin(\frac{x+y}{2})\cos(\frac{x-y}{2})$$ $$\cos x-\cos y = -2\sin(\frac{x+y}{2})\sin(\frac{x-y}{2})$$ Now substitute these formulas into the identity: $$\frac{2\sin(\frac{x+y}{2})\cos(\frac{x-y}{2})}{-2\sin(\frac{x+y}{2})\sin(\frac{x-y}{2})} =-\frac{\cos(\frac{x-y}{2})}{\sin(\frac{x-y}{2})}$$

Step 3: Simplify the equation

Notice that we can simplify some factors: $$\frac{\cos(\frac{x-y}{2})}{\sin(\frac{x-y}{2})}=-\frac{\cos(\frac{x-y}{2})}{\sin(\frac{x-y}{2})}$$ Since both sides of the equation are equal, we have successfully proven the identity: $$\frac{\sin x+\sin y}{\cos x-\cos y}=-\cot \left(\frac{x-y}{2}\right)$$

Key Concepts

Cotangent Function

The cotangent function is one of the six primary trigonometric functions and is the reciprocal of the tangent function. Represented as \( \text{cot} \), it's defined as the ratio of the adjacent side to the opposite side in a right triangle, or in terms of sine and cosine functions, as \( \text{cot}(θ) = \frac{\text{cos} (θ)}{\text{sin} (θ)} \). Understanding cotangent is crucial when simplifying trigonometric expressions, as it interrelates with other trigonometric functions, providing pathways to manipulate and rewrite equations.

In the exercise you're working on, the cotangent function is expressed in terms of cosine and sine for the angle \( \frac{x-y}{2} \), which then allows the use of trigonometric identities to prove the given identity. It's important to be comfortable with expressing cotangent this way because doing so unlocks the ability to apply other mathematical techniques, such as the sum-to-product formulas.

Sum-to-Product Formulas

Sum-to-product formulas are a set of trigonometric identities that transform the sum or difference of trigonometric functions into products of functions. These formulas are handy for integrating trigonometric functions and for proving other trigonometric identities. The sum-to-product formulas for the sine function, for instance, are:

\[ \text{sin}(x) + \text{sin}(y) = 2 \text{sin}\bigg(\frac{x+y}{2}\bigg)\text{cos}\bigg(\frac{x-y}{2}\bigg) \]
\[ \text{sin}(x) - \text{sin}(y) = 2 \text{cos}\bigg(\frac{x+y}{2}\bigg)\text{sin}\bigg(\frac{x-y}{2}\bigg) \]

Applying these allows us to rework the left-hand side of the trigonometric identity you have into a form that can be directly compared with the cotangent function on the right-hand side. This highlights the essence of these formulas: converting an equation into a more manageable form that can be easily simplified or compared with other expressions.

Trigonometric Simplification

Trigonometric simplification is the process of reducing trigonometric expressions to their simplest form, often using trigonometric identities. Simplification is useful in solving equations, proving identities, and finding function values. For instance, when you see complex trigonometric expressions, look for common factors, reciprocal identities, Pythagorean identities, or angle-sum formulas to simplify.

In the provided exercise, once the sum-to-product formulas are applied, we observe common factors in the numerator and denominator of the resulting expression. By canceling out the common terms, we simplify the expression to match both sides of the identity, proving it to be true. The art of simplification lies in recognizing patterns and remembering key identities that can transform an expression into a neater and more comprehensible form, just as we did when simplifying \( -\text{cot} \) to exemplify equivalence.