Problem 72 - Algebra and Trigonometry

Question

59–76 Prove the identity. $$\frac{\sin 3 x+\sin 7 x}{\cos 3 x-\cos 7 x}=\cot 2 x$$

Step-by-Step Solution

Answer

The identity is \( \frac{\sin 3x + \sin 7x}{\cos 3x - \cos 7x} = \cot 2x \).

Step 1: Recall Relevant Identities

To solve this problem, we'll need a few trigonometric identities. The sum-to-product identities are crucial here: 1. \( \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right)\cos\left(\frac{A - B}{2}\right) \)2. \( \cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right)\sin\left(\frac{A - B}{2}\right) \)

Step 2: Apply Sum-to-Product Identities to the Numerator

For the numerator, \( \sin 3x + \sin 7x \), apply the sum-to-product identity:\[\sin 3x + \sin 7x = 2 \sin\left(\frac{3x + 7x}{2}\right)\cos\left(\frac{3x - 7x}{2}\right) = 2 \sin(5x)\cos(-2x)\]Recall that \( \cos(-2x) = \cos(2x) \).

Step 3: Apply Sum-to-Product Identities to the Denominator

For the denominator, \( \cos 3x - \cos 7x \), apply the difference-to-product identity:\[\cos 3x - \cos 7x = -2 \sin\left(\frac{3x + 7x}{2}\right)\sin\left(\frac{3x - 7x}{2}\right) = -2 \sin(5x)\sin(-2x)\]Note that \( \sin(-2x) = -\sin(2x) \), so the expression becomes:\[-2 \sin(5x)(-\sin(2x)) = 2 \sin(5x)\sin(2x)\]

Step 4: Simplify the Fraction

Substitute the results from Steps 2 and 3 into the given expression:\[\frac{2 \sin(5x)\cos(2x)}{2 \sin(5x)\sin(2x)}\]The \(2 \sin(5x)\) terms cancel out, leaving:\[\frac{\cos(2x)}{\sin(2x)}\]

Step 5: Rewrite the Result Using Trigonometric Definitions

Recognize that \( \frac{\cos(2x)}{\sin(2x)} = \cot(2x) \). Therefore, the original expression simplifies to \( \cot(2x) \). This proves the identity.

Key Concepts

Sum-to-Product Identities

When dealing with expressions involving sums or differences of sine and cosine, the sum-to-product identities can be a powerful tool for simplification. These identities help us rewrite a sum or difference of trigonometric functions (like sine or cosine) as a product, making it easier to simplify complex expressions.

Consider the sum-to-product identity for sine:

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

This identity is useful when you see an expression like \( \sin 3x + \sin 7x \). By applying it, the expression is transformed from a sum into a product, which can simplify the problem.

For cosine, there's a similar identity:

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

This identity is helpful in cases such as \( \cos 3x - \cos 7x \), turning a difference into a product and thereby making further simplification possible.

Using these identities helps in breaking down complicated trigonometric expressions into something more manageable. This not only makes it easier to solve and simplify but is also crucial in proving identities.

Trigonometric Proofs

Trigonometric proofs often require a solid understanding of various identities, properties, and transformations. These proofs are essentially about showing that two expressions are equivalent by using trigonometric rules and identities.

In our example problem, the task was to prove the identity \( \frac{\sin 3x + \sin 7x}{\cos 3x - \cos 7x} = \cot 2x \). This involves a series of steps where we use identities to transform the left side of this equation into the right side.

Start with applying the appropriate sum-to-product identities to separate and combine the terms effectively. Once the expression is transformed, further simplification involves cancellation of common terms, leading us to the target expression.

Proofs like these require:

• Choosing the right identities or theorems to apply.
• Careful arithmetic manipulation.
• Verification of each step to ensure accuracy.

By following these steps, you'll not only prove the identity but also deepen your understanding of trigonometric relationships. It's about recognizing patterns and making strategic transformations to reveal hidden equivalences.

Cotangent Function

The cotangent function, denoted as \( \cot \), is closely related to the tangent function. It is the reciprocal of tangent, defined as the ratio of the adjacent side to the opposite side in a right triangle, or algebraically:

• \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \)

Understanding this relationship is crucial especially when dealing with proofs and transformations where the cotangent appears, like in our example proof. The task was to show that a fraction of trigonometric sums and differences equals \( \cot 2x \).

Knowing that \( \cot(2x) = \frac{\cos(2x)}{\sin(2x)} \) helps us recognize when we've successfully simplified an expression. Often in trigonometric proofs, the goal is to transform a more complex expression into a simpler form, such as a single function like cotangent.

The cotangent function also appears frequently in various trigonometric identities and equations, making it an essential part of trigonometry. By mastering the concept of cotangent, we can more easily handle a wide variety of trigonometric problems and proofs.