Proving Identities Prove the identity. cos(x+y) cos(x-y)=cos^2 x-sin^2 y

Name: Problem 43, Chapter 7: Precalculus Mathematics for Calculus
Uploaded: 2020-05-20T18:23:07-08:00
Duration: 5 min 8 s
Channel: Caelan Thomas
Description: Proving Identities Prove the identity. cos(x+y) cos(x-y)=cos^2 x-sin^2 y

step 1 We want to prove the identity, cosine x plus y times cosine x minus y equals cosine squared x mi

Proving Identities Prove the identity. cos(x+y)...

Key Concepts

Cosine Sum and Difference Formulas

These formulas, which state that cos(x+y) = cos x cos y ? sin x sin y and cos(x?y) = cos x cos y + sin x sin y, are key to breaking down compound angle expressions into products of simple trigonometric functions. They allow one to express the product cos(x+y) cos(x?y) in terms of products of cosine and sine functions of individual angles.

Difference of Squares Factorization

This algebraic concept is used when multiplying the expressions obtained from the cosine sum and difference formulas. Recognizing that the product (A ? B)(A + B) can be written as A² ? B² simplifies the expression significantly, reducing the calculation and paving the way for further simplification.

Pythagorean Identity

The Pythagorean identity, which states that sin²? + cos²? = 1, is often employed in trigonometric proofs to simplify and substitute expressions. In this proof, it helps in combining terms after factorization so that the final expression can be neatly rewritten in terms of cos²x and sin²y.

Algebraic Manipulation in Trigonometric Proofs

Beyond the specific trigonometric identities, the proof relies on careful algebraic manipulation—expanding products, combining like terms, and recognizing patterns—to transform an initially complex expression into a simpler equivalent form. This process is fundamental in proving and understanding trigonometric identities.

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