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Chapter 20: Problem 26

Prove the given identities. $$\cos (x+y) \cos (x-y)=\cos ^{2} x-\sin ^{2} y$$

Short Answer

Expert verified
The identity is proven: \( \cos(x+y) \cos(x-y) = \cos^2 x - \sin^2 y \).

Step by step solution

01

Expand the Left Side

We start by expanding the expression \( \cos(x+y) \cos(x-y) \). Using the product-to-sum identities, we get:\[\cos(x+y) \cos(x-y) = \frac{1}{2} \left[\cos((x+y) + (x-y)) + \cos((x+y) - (x-y))\right]\]This simplifies to:\[\frac{1}{2} [\cos(2x) + \cos(2y)]\]
02

Simplify Using Double Angle Formulas

Now, apply the double angle formulas to simplify the expressions:\[\cos(2x) = 2\cos^2 x - 1\]\[\cos(2y) = 2\cos^2 y - 1\]Substitute these into the simplified left side:\[\frac{1}{2} [2\cos^2 x - 1 + 2\cos^2 y - 1]\]This becomes:\[\cos^2 x + \cos^2 y - 1\]
03

Use Trigonometric Identity

Recall the Pythagorean identity, which states:\[\sin^2 y + \cos^2 y = 1\]Thus, \( \cos^2 y = 1 - \sin^2 y \). Substitute this into the expression:\[\cos^2 x + (1 - \sin^2 y) - 1\]Simplifying gives:\[\cos^2 x - \sin^2 y\]
04

Verify Both Sides

Both sides of the equation now match:\[\cos(x+y) \cos(x-y) = \cos^2 x - \sin^2 y\]Therefore, the identity is verified as true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Addition Formula
The cosine addition formula is a fundamental trigonometric identity. It is used to break down complex trigonometric expressions involving the sum or difference of two angles. This formula is expressed as:
  • \(\cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b)\)
  • \(\cos(a - b) = \cos(a) \cos(b) + \sin(a) \sin(b)\)
These identities help simplify expressions by relating them to single angle trigonometric functions. Breaking down a complex expression like \(\cos(x+y) \cos(x-y)\) becomes manageable using these formulas. In our problem, leveraging this identity was a crucial step to transform the terms into a more convenient form for further simplification.
Product-to-Sum Identities
Product-to-sum identities are powerful tools in trigonometry. They transform products of sine and cosine functions into sums, which are generally easier to handle. For cosine, one formula is:
  • \(\cos(a)\cos(b) = \frac{1}{2} [\cos(a+b) + \cos(a-b)]\)
This identity was key in our exercise to expand \(\cos(x+y) \cos(x-y)\). By converting a product into a sum, these identities simplify the expression, making it easier to apply other identities like double angle formulas. They essentially act as a transformative bridge to enable deeper simplification of trigonometric expressions.
Double Angle Formulas
Double angle formulas simplify expressions involving double angles (such as \(2x\) or \(2y\)). The formula for cosine is:
  • \(\cos(2a) = 2\cos^2(a) - 1\)
This formula was used in our solution to simplify \(\cos(2x)\) and \(\cos(2y)\) when the problem was expanded using product-to-sum identities. They allow complex compositions of angles to be expressed with single angle functions, thus making it easier to apply other trigonometric identities for further simplification.
Pythagorean Identity
The Pythagorean identity is a fundamental relation in trigonometry. It relates the squares of sine and cosine functions:
  • \(\sin^2(a) + \cos^2(a) = 1\)
In our problem, this identity was utilized to replace \(\cos^2(y)\) with \(1 - \sin^2(y)\), simplifying the equation to \(\cos^2(x) - \sin^2(y)\). This identity ensures that one can eliminate one trigonometric function by expressing it in terms of another, ultimately facilitating the matching of the left and right sides of the equation to prove the identity.

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