Problem 81 - Contemporary Precalculus

Question

Prove the identity. $$\frac{\cos x-\sin y}{\cos y-\sin x}=\frac{\cos y+\sin x}{\cos x+\sin y}$$

Step-by-Step Solution

Answer

Question: Prove that the given trigonometric identity holds true: $$ \frac{\cos x-\sin y}{\cos y-\sin x}=\frac{\cos y+\sin x}{\cos x+\sin y} $$ Answer: Proven through a series of algebraic manipulations and trigonometric identities, the given trigonometric identity holds true. See the provided step-by-step solution for detailed explanation.

Step 1: Recognition of symmetry

Note that both sides of the equation are symmetric, meaning that they have the same structure but the x and y are switched between the numerator and denominator for both sides. We will use this symmetry to prove the identity.

Step 2: Cross-multiplication

We will start by making a single fraction on both sides of the equation, which will allow for easier manipulation. To do this, cross-multiply the two fractions being equal, as follows: $$ (\cos x - \sin y)(\cos x + \sin y) = (\cos y - \sin x)(\cos y + \sin x) $$

Step 3: Expand both sides of the equation

Use the distributive property to expand and simplify both sides of the equation: $$ (\cos^2 x - \sin^2 y) = (\cos^2 y - \sin^2 x) $$

Step 4: Use the Pythagorean identity

Recall the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\). We can use this identity to replace the squares of sine and cosine functions in the equation with their respective complements: $$ (1 - \sin^2 x - \sin^2 y) = (1 - \sin^2 y - \sin^2 x) $$

Step 5: Simplify and rearrange

From the last equation, we have: $$ 1 - \sin^2 x - \sin^2 y = 1 - \sin^2 y - \sin^2 x $$ Thus, we've proven that the two sides of the original equation are equal, so the given trigonometric identity holds true: $$ \frac{\cos x-\sin y}{\cos y-\sin x}=\frac{\cos y+\sin x}{\cos x+\sin y} $$

Key Concepts

Symmetry in Equations

A fundamental concept in mathematics and particularly in understanding trigonometric identities is recognizing symmetry in equations. When we address symmetry, we are often referring to a balanced and proportional similarity that maintains the same characteristics even if certain elements are interchanged. The identity we are proving here displays a kind of symmetry where the roles of variables, namely x and y, can be interchanged without affecting the equality. This is a powerful observation that simplifies the process of proving identities. By exploiting this symmetry, we impose a methodological approach that could reveal underlying relationships within the trigonometric functions themselves.

When dealing with the symmetry in trigonometric identities, it is crucial to be keenly observant, as this can lead to straightforward computational pathways. Symmetry enables us to transform complex problems into simpler forms, often revealing a direct path to the solution. In the case of the given exercise, recognizing the symmetry allows us to cross-multiply confidently, knowing that the structure of both sides of the equation is consistent even as the roles of the variables are exchanged.

Cross-Multiplication

Cross-multiplication is a widely used technique in algebra that simplifies the process of dealing with fractions and proving identities. This method is especially useful when you are faced with an equation consisting of two fractions set equal to each other. The process involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa, effectively eliminating the fraction format and leading to a simpler equation to handle.

In our exercise, cross-multiplying both sides of the equation consolidates the relationship between the trigonometric functions, transforming the identity into an expressible format that can be conveniently manipulated further. The beauty of cross-multiplication lies in its ability to clear the fractions and provides a groundwork for applying trigonometric identities, like the Pythagorean identity, thus greatly aiding in the simplification of complex trigonometric equations.

Pythagorean Identity

The Pythagorean identity is a pillar of trigonometry, foundational for understanding how the sine and cosine functions relate to one another. It is one of several trigonometric identities that are derived from the Pythagorean theorem and it states that for any angle \theta, the equation \(\sin^2(\theta) + \cos^2(\theta) = 1\) holds true. This relationship is a direct result of the inherent properties of right-angled triangles and the unit circle.

When we apply this identity, as shown in the textbook solution, we are able to replace \(\cos^2 x\) and \(\cos^2 y\) with their Pythagorean complements \(1 - \sin^2 x\) and \(1 - \sin^2 y\), respectively. This is a crucial step in proving trigonometric identities, as it often allows us to convert the equation into a form that can be more readily simplified or compared. Recognizing when and how to apply the Pythagorean identity can dramatically reduce the complexity of proving an identity, transforming it into a manageable endeavor. Understanding and mastering the use of this identity is essential for anyone studying trigonometry.