Problem 37 - Algebra and Trigonometry

Question

\(23-40\) . Prove the identity. $$ \frac{\sin (x+y)-\sin (x-y)}{\cos (x+y)+\cos (x-y)}=\tan y $$

Step-by-Step Solution

Answer

The identity is proven: it simplifies to \( \tan y \).

Step 1: Understand the Identity

The expression given is \( \frac{\sin(x+y) - \sin(x-y)}{\cos(x+y) + \cos(x-y)} \). We need to prove that this is equal to \( \tan y \).

Step 2: Apply Sum-to-Product Formulas for Sine

The sum-to-product identity for sine is \( \sin A - \sin B = 2 \cos \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right) \). Apply this to \( \sin(x+y) - \sin(x-y) \).

Step 3: Simplify the Numerator

Apply the formula: \( \sin(x+y) - \sin(x-y) = 2 \cos(x) \sin(y) \). This gives us a simplified numerator of \( 2 \cos(x) \sin(y) \).

Step 4: Apply Sum-to-Product Formulas for Cosine

The sum-to-product identity for cosine is \( \cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \). Apply this to \( \cos(x+y) + \cos(x-y) \).

Step 5: Simplify the Denominator

Apply the formula: \( \cos(x+y) + \cos(x-y) = 2 \cos(x) \cos(y) \). This gives us a simplified denominator of \( 2 \cos(x) \cos(y) \).

Step 6: Simplify the Expression

The expression becomes \( \frac{2 \cos(x) \sin(y)}{2 \cos(x) \cos(y)} \). The factor \( 2 \cos(x) \) cancels from the numerator and denominator.

Step 7: State the Result

The expression simplifies to \( \frac{\sin(y)}{\cos(y)} \), which is \( \tan(y) \). Thus, the given identity is proven.

Key Concepts

Sum-to-Product Formulas

To understand how the sum-to-product formulas help in proving trigonometric identities, let's dive into what these formulas are. These formulas are essential tools in trigonometry, allowing us to transform sums or differences of trigonometric functions into products. This can often simplify complex expressions, making them easier to work with.

For sine functions, the sum-to-product formula is expressed as:

• \( \sin A - \sin B = 2 \cos \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right) \)

This formula helps us change the difference between two sine terms into a product that is often more manageable. In the context of the given exercise, replacing the difference \( \sin(x+y) - \sin(x-y) \) with its product form substantially simplifies the overall expression.

Similarly, for cosine functions, the sum-to-product formula is:

• \( \cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \)

Applying this to the sum \( \cos(x+y) + \cos(x-y) \) similarly assists in simplifying our expression. Through these transformations, we pave the way to solve trigonometric identities more efficiently.

Sine and Cosine Identities

Sine and cosine identities form the backbone of trigonometric equations and transformations. Understanding these identities is crucial for unraveling complex expressions and proving identities.

Consider the basic identities involving sine and cosine:

• \( \sin^{2} \theta + \cos^{2} \theta = 1 \)
• Common angle sum identities include: \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \) and \( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \)

These identities are the building blocks for manipulating and simplifying trigonometric expressions.

In the given exercise, the identities are applied after transforming the expression using sum-to-product formulas. This allows us to simplify the expression to a point where cancelling similar terms makes proving the identity a straightforward task. Such manipulation is typical of working with identities and essential for students to understand.

Proving Identities

The art of proving identities in trigonometry involves strategically transforming and simplifying an expression to reveal its true nature. The process can seem daunting at first, yet it becomes significantly easier with practice and close attention to trigonometric laws.

Here are some key steps when embarking on proving identities:

• Identify known identities and use them for substitution or transformation. Familiarizing with basic and compound identities is immensely helpful.
• Apply algebraic techniques such as factoring, expanding, and cancelling common terms.
• Stay clear of assumptions; always work from one side of the equation toward the other rather than assuming they are already equal.

In the exercise, we demonstrated proving the identity \( \frac{\sin(x+y) - \sin(x-y)}{\cos(x+y) + \cos(x-y)} = \tan y \). By applying sum-to-product formulas and simplifying the terms step-by-step, the simplicity of \( \frac{\sin(y)}{\cos(y)} = \tan(y) \) emerges at the final step. This methodical approach showcases how difficult identities can be proven true by breaking them down into understandable parts.