Problem 63 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

Question: Prove the identity $$\cos (x+y) \cos (x-y)=\cos ^{2} x \cos ^{2} y-\sin ^{2} x \sin ^{2} y$$ Solution: 1. Start with the given identity: \(\cos (x+y) \cos (x-y)=\cos ^{2} x \cos ^{2} y-\sin ^{2} x \sin ^{2} y\) 2. Expand both \(\cos (x + y)\) and \(\cos (x - y)\) using the sum and difference identities: \([\cos x \cos y - \sin x \sin y][\cos x \cos y + \sin x \sin y]\) 3. Multiply the two expressions on the LHS: \(\cos ^2 x \cos ^2 y + \sin ^2 x \sin ^2 y - \sin ^2 x \cos ^2 y - \sin ^2 y \cos ^2 x\) 4. Use the Pythagorean identity and rearrange the terms: \(\cos^2 x \cos^2 y - \sin^2 x \sin^2 y\) 5. Both sides of the equation now match, hence the identity is proved. Identity proved: $$\cos (x+y) \cos (x-y)=\cos ^{2} x \cos ^{2} y-\sin ^{2} x \sin ^{2} y$$

Step 1: Write down the given identity

We have, $$\cos (x+y) \cos (x-y)=\cos ^{2} x \cos ^{2} y-\sin ^{2} x \sin ^{2} y$$

Step 2: Expand both \(\cos (x + y)\) and \(\cos (x - y)\) using the sum and difference identities

Apply the identities to the LHS of the equation: $$[\cos x \cos y - \sin x \sin y][\cos x \cos y + \sin x \sin y]$$

Step 3: Multiply the two expressions on the LHS

Using the distributive property, we get: $$(\cos x \cos y - \sin x \sin y)(\cos x \cos y + \sin x \sin y) = \cos ^2 x \cos ^2 y + \sin ^2 x \sin ^2 y - \sin ^2 x \cos ^2 y - \sin ^2 y \cos ^2 x$$

Step 4: Use the Pythagorean identity and rearrange the terms

The Pythagorean identity states that \(\sin^2 x + \cos^2 x = 1\). So, let's rearrange the expression from Step 3: $$\cos ^2 x \cos ^2 y + \sin ^2 x \sin ^2 y - \sin ^2 x \cos ^2 y - \sin ^2 y \cos ^2 x = \cos^2 x \cos^2 y - \sin^2 x \sin^2 y$$

Step 5: Proving the identity

Since both sides of the equation match now, the identity is proved: $$\cos (x+y) \cos (x-y)=\cos ^{2} x \cos ^{2} y-\sin ^{2} x \sin ^{2} y$$

Key Concepts

Sum and Difference Identities

The sum and difference identities are crucial tools in trigonometry that help to simplify complex expressions. These identities show how the trigonometric functions of sum or difference of two angles can be expressed in terms of functions of individual angles. They are given by:

• \(\cos (x + y) = \cos x \cos y - \sin x \sin y\)
• \(\cos (x - y) = \cos x \cos y + \sin x \sin y\)

In our exercise, we use these identities to expand \(\cos(x+y)\) and \(\cos(x-y)\) as a first step to simplifying the original expression. This expansion is vital as it transforms the product of cosines into a manageable algebraic expression. Knowing these identities allows us to rewrite and manipulate trigonometric functions more easily. As you practice, try to remember these formulas by relating them to visual aids like the unit circle.

Distributive Property

The distributive property is a fundamental mathematical principle that states how multiplication is distributed over addition. Often expressed as \(a(b+c) = ab + ac\), it plays a key role in expanding algebraic expressions. In our trigonometric identity problem, the distributive property is used to carry out the multiplication between the two expanded trigonometric expressions:

• First term: \( (\cos x \cos y - \sin x \sin y)(\cos x \cos y + \sin x \sin y) \)
• Distribute to get four components resulting from the multiplication.

This gives rise to a series of products which are then carefully collected and rearranged. Practically, it enables us to break complex trigonometric products into simpler components, making the solving process much smoother. Being comfortable with this property makes more advanced operations—like factorization—a breeze.

Pythagorean Identity

The Pythagorean identity is perhaps the most famous identity in trigonometry. It asserts that for any angle \(x\), the relationship \(\sin^2 x + \cos^2 x = 1\) holds true. This property arises from the equation of a unit circle, where the radius is 1. Here is how the identity is utilized:

• By recognizing and applying this identity, terms such as \((\cos^2 x \cos^2 y - \sin^2 x \sin^2 y)\) reduce gracefully.
• Assists in combining similar terms effectively, rendering the equation valid and proved.

In our exercise, the use of this identity is the cornerstone for bringing our solution together. Whenever you encounter trigonometric expressions involving squares of sine and cosine, remember this identity. It simplifies expressions and aids in verifying identities by making the relationships between sine and cosine straightforward and intuitive.