Problem 28 - Precalculus Essentials

Question

Verify each identity. $$\frac{\sin x+\sin y}{\sin x-\sin y}=\tan \frac{x+y}{2} \cot \frac{x-y}{2}$$

Step-by-Step Solution

Answer

The identity \(\frac{\sin x+\sin y}{\sin x-\sin y} = \tan \frac{x+y}{2} \cot \frac{x-y}{2}\) is verified when both sides are reduced to the simplified form of \(\frac{\sin(x+y)}{\sin(x-y)}\).

Step 1: Apply the sum and difference identity to the right side

The right side of the equation, \(\tan \frac{x+y}{2} \cot \frac{x-y}{2}\), can be rewritten using the sum and difference identities. Express these as \(\tan\left(\frac{x}{2}+\frac{y}{2}\right)\cot\left(\frac{x}{2}-\frac{y}{2}\right)\). By applying the sum and difference formula for tangent, obtain \(\frac{\sin(x+y)}{\cos(x+y)}\cdot\frac{\cos(x-y)}{\sin(x-y)}\).

Step 2: Simplify Right Side

The right side of the equation simplifies to \(\frac{\sin(x+y)}{\sin(x-y)}\). This is derived by cancelling out the common term, \(\cos(x+y)\), from both the numerator and the denominator.

Step 3: Apply the sum and difference identity for sine to the left side

The left side of the equation, \(\frac{\sin x+\sin y}{\sin x-\sin y}\), could be rewritten using the sum and difference identities. This simplifies to \(\frac{\sin(x+y)}{\sin(x-y)}\).

Step 4: Compare Both Sides

Both sides now become equal. This confirms that the trigonometric identity is indeed verified.

Key Concepts

Sum and Difference Identity

Trigonometric identities help us connect various trigonometric functions and simplify complex expressions. The sum and difference identities are particularly useful in manipulating expressions involving angles. They offer:

• An easy way to express trigonometric functions of added or subtracted angles.
• Simplified forms that reveal interesting properties of the functions.

For example, the sum identity for sine is given by:\[\sin(x + y) = \sin x \cos y + \cos x \sin y\]And for the difference, it's:\[\sin(x - y) = \sin x \cos y - \cos x \sin y\]
These identities are the foundation for rewriting expressions like the one in the exercise. Through these identities, complex expressions transform into simplified forms, enabling easier verification of trigonometric identities or solutions in equations.

Tangent and Cotangent

Tangent and cotangent are two fundamental trigonometric functions. They are the ratios of sine and cosine functions, with distinct properties:

• The tangent function, \( \tan x = \frac{\sin x}{\cos x} \), reveals the slope of the angle.
• Cotangent, \( \cot x = \frac{\cos x}{\sin x} \), is the reciprocal of tangent.

In the provided exercise, we utilize these functions in their halved form:\[\tan \left( \frac{x+y}{2} \right) \quad \text{and} \quad \cot \left( \frac{x-y}{2} \right)\]
This implies evaluating \'angle half-sum\' and \'angle half-difference\'. These are essential for understanding transformations and simplifying trigonometric functions. By leveraging these identities, we can compare sides in an equation, showing their equivalence with less complexity.

Simplifying Trigonometric Expressions

Working with trigonometric expressions can initially seem daunting, but simplification plays a crucial role. At its core, the simplified form of a trigonometric expression may:

• Reduce complexity and reveal insightful properties.
• Make it easier to solve or verify equations.

Simplifying entails applying known identities to reformulate expressions without altering their value. In our case:

• The left side \( \frac{\sin x + \sin y}{\sin x - \sin y} \) is simplified using the sine sum and difference identities.
• The right side \( \tan \frac{x+y}{2} \cot \frac{x-y}{2} \) uses halved-angle forms to transform into a simpler result.

Once both sides are simplified similarly, demonstrating their equality closes the loop on verifying the identity. This underscores the elegance of trigonometry—where seemingly complex expressions unify under identities to reveal their true connections.