Proving the Trigonometric Identity: sin x sin y 1/2 [cos(x - y) - cos(x y)]

Introduction

Understanding and proving trigonometric identities is a fundamental skill in mathematics, especially in fields such as physics, engineering, and advanced calculus. One common identity that often arises in applications is the relationship between the product of the sine of two angles, and a combination of cosine functions: sin x sin y 1/2 [cos(x - y) - cos(x y)]. In this article, we will explore a straightforward proof of this identity. No advanced courses are required, just a bit of basic trigonometry and an understanding of sum and difference identities.

Prerequisites

To follow along with this proof, it is recommended to have a basic understanding of the following concepts:

Pythagorean theorem: x2 y2 z2 Sine and cosine definitions: sine opposite / hypotenuse, cosine adjacent / hypotenuse Sum and difference identities for cosine

The Proof

Let's start by recalling the sum and difference identities for cosine, and then we will use these to derive the desired identity.

Sum and Difference Identities for Cosine

The sum and difference identities for cosine are:

cos (x y) cos x cos y - sin x sin y cos (x - y) cos x cos y sin x sin y

Deriving the Identity

From the above identities, we can isolate the product of the sine functions:

cos (x - y) cos x cos y sin x sin y

cos (x y) cos x cos y - sin x sin y

Subtract the second equation from the first:

cos (x - y) - cos (x y) (cos x cos y sin x sin y) - (cos x cos y - sin x sin y)

Simplify the right side:

cos (x - y) - cos (x y) 2 sin x sin y

Divide both sides by 2:

sin x sin y 1/2 [cos (x - y) - cos (x y)]

Verification and Application

This identity can be verified by substituting specific values for x and y and checking the results. For example, if x y 30° or π/6 radians, both sides of the equation are equal to 1/4.

Understanding this identity is beneficial in various applications. For instance, in signal processing and electrical engineering, it is often used to analyze the interactions between different sinusoidal signals.

Conclusion

Proving the identity sin x sin y 1/2 [cos(x - y) - cos(x y)] involves simple algebraic manipulation and the application of sum and difference identities for cosine. With a basic understanding of trigonometric principles, anyone can derive and verify this identity. Practice with such identities not only deepens one's understanding of trigonometry but also enhances problem-solving skills in mathematics and related fields.