Problem 88 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

Based on the given step by step solution, the short answer is: We proved that \(\frac{\sin x-\sin y}{\cos x+\cos y} = \tan\left(\frac{x-y}{2}\right)\) by first applying the sum-to-product formulas to both the numerator and denominator, then simplifying the expression, and finally identifying the form of the tangent function.

Step 1: Apply sum-to-product formulas

Let's express the numerator, \(\sin x - \sin y\) using the sum-to-product formula: $$\sin x - \sin y = 2\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)$$ Similarly, express the denominator, \(\cos x + \cos y\) using the sum-to-product formula: $$\cos x + \cos y = 2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$$

Step 2: Substitute the expressions

Now substitute these expressions in the given identity: $$\frac{\sin x-\sin y}{\cos x+\cos y} = \frac{2\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)}{2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)}$$

Step 3: Simplify the expression

Observe that the \(2\cos\left(\frac{x+y}{2}\right)\) in the numerator and denominator cancels out: $$\frac{\sin x-\sin y}{\cos x+\cos y} = \frac{\sin\left(\frac{x-y}{2}\right)}{\cos\left(\frac{x-y}{2}\right)}$$

Step 4: Applying the definition of tangent

Notice that the expression now has the form of tangent: $$\frac{\sin x-\sin y}{\cos x+\cos y} = \tan\left(\frac{x-y}{2}\right)$$ Since we have successfully shown that the given expression is equal to the identity, the proof is complete.

Key Concepts

Sum-to-Product Formulas

Sum-to-product formulas are incredibly useful in trigonometry. They allow us to transform sums or differences of trigonometric functions into products, which often simplifies the calculation or proof of trigonometric identities. These formulas are built on fundamental trigonometric identities, and understanding them can be a great help in solving complex problems.

Here's how they work:

• For the sine function, the formula is: \( \sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \)
• For the cosine function, the formula is: \( \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \)

These formulas take advantage of symmetry in the cosine and sine functions, giving us a different way to express combinations of angles. In our problem, we used the sum-to-product formulas to transform the numerator and denominator separately. By simplifying these terms, it made the proof more straightforward.

Trigonometric Proofs

Trigonometric proofs are a way of verifying that a trigonometric identity holds true for all values within its domain. Proofs motivate the understanding of identities by breaking them down, checking step by step why each part of the identity works.

When performing a trigonometric proof, like the one in our problem, we typically do the following:

• Start by setting up what we know. This means writing down the identity to prove.
• Use known identities, like sum-to-product, to transform the expression. This can involve rewriting parts of the expression using these identities.
• Simplify! Cancel out and reduce terms wherever possible, as we did by canceling the common terms in the numerator and denominator.
• Conclude by checking if both sides of the identity are equivalent.

Through logical reasoning and manipulation of formulas, trigonometric proofs teach us to see connections between different trigonometric concepts. This is why understanding each formula's role in a proof is critical.

Tangent Function

The tangent function, commonly denoted as \( \tan \), is one of the primary trigonometric functions and is crucial in solving various trigonometric problems. It is defined as the ratio of the sine and cosine functions: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

In our problem, after simplifying the expression using trigonometric identities, we end up with a fraction whose numerator is a sine function and whose denominator is a cosine function. Thus, it resembles the tangent function:

\[ \frac{\sin\left(\frac{x-y}{2}\right)}{\cos\left(\frac{x-y}{2}\right)} = \tan\left(\frac{x-y}{2}\right) \]

Understanding the tangent function's properties helps in many scenarios. For example:

• It is undefined whenever the cosine is zero since division by zero is not allowed.
• The function is periodic, with a period of \( \pi \), which means it repeats itself every \( \pi \) units.

The tangent function's behavior is essential when working with right-angle triangles or when modeling periodic phenomena. Recognizing this in the context of proofs helps you understand how parts of an equation fit together.