1. What is the Product to Sum Identity?

The product-to-sum identities are trigonometric identities that convert products of trigonometric functions into sums or differences. This particular identity converts the product of two sine functions into a difference of cosine functions.

2. How Does the Identity Work?

The identity is expressed as:

Derivation: This identity comes from adding and subtracting the cosine addition formulas:

• \(\cos(A+B) = \cos A \cos B - \sin A \sin B\)
• \(\cos(A-B) = \cos A \cos B + \sin A \sin B\)

Subtracting these gives: \(\cos(A-B) - \cos(A+B) = 2\sin A \sin B\)

3. Importance of the Identity

Applications: This identity is particularly useful in:

• Simplifying trigonometric expressions
• Solving integrals involving products of trigonometric functions
• Analyzing wave interference patterns in physics
• Signal processing and Fourier analysis

4. Using the Calculator

Instructions: Enter two angles (A and B) in degrees. The calculator will show both sides of the identity, demonstrating their equality.

5. Frequently Asked Questions (FAQ)

Q1: Are there other product-to-sum identities?
A: Yes, similar identities exist for sin A cos B and cos A cos B products.

Q2: Why is this identity useful in calculus?
A: It helps integrate products of trigonometric functions by converting them into sums.

Q3: Does this work for any angle values?
A: Yes, the identity holds for all real values of A and B.

Q4: Can this be used for complex numbers?
A: Yes, with appropriate definitions of trigonometric functions for complex arguments.

Q5: How precise are the calculations?
A: The calculator shows results rounded to 6 decimal places for clarity.