Problem 67 - Precalculus with Limits

Question

In Exercises 61 - 70, prove the identity. \( \cos(x + y) \cos(x - y) = \cos^2 x - \sin^2 y \)

Step-by-Step Solution

Answer

The original identity \( \cos(x + y) \cos(x - y) = \cos^2 x - \sin^2 y \) is indeed correct.

Step 1: Expand the left side

The left side of the equation involves the cosine of the sum and difference of two angles. Using the sum-to-product identities, the cosine of a sum or difference can be rewritten as follows: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \) and \( \cos(a - b) = \cos a \cos b + \sin a \sin b \) respectively. Therefore, \( \cos(x + y) \) and \( \cos(x - y) \) can be expanded using these identities and then multiplied.

Step 2: Multiplication

Substitute these expansions into the original equation to get: \( [\cos x \cos y - \sin x \sin y] [\cos x \cos y + \sin x \sin y] \). You now have to multiply these two factors. Use the well-known formula for the product (a-b)(a+b) = \( a^2 - b^2 \), thus obtaining: \( \cos^2 x \cos^2 y - \sin^2 x \sin^2 y \).

Step 3: Apply trigonometric identity

Now, you proceed to convert the terms involving sine using the Pythagorean Identity \( \cos^2 a + \sin^2 a = 1 \) from which it follows that \( \sin^2 a = 1 - \cos^2 a \). Apply this conversion to the current equation, replacing \( \sin^2 x = 1 - \cos^2 x \) and \( \sin^2 y = 1 - \cos^2 y \). Consequently, you obtain the equation: \( \cos^2 x \cos^2 y - (1 - \cos^2 x)(1 - \cos^2 y) \). However, the right side of the initial equation contains terms independent on x and y. Consider that \( \cos^2 y \) in the obtained equation can be also represented as \( 1 - \sin^2 y \), and substitute it.

Step 4: Simplification

After simplifying the equation from step 3, finally you can find: \( \cos^2 x - \sin^2 y \), which is the required form of the given identity. Thus, the original identity \( \cos(x + y) \cos(x - y) = \cos^2 x - \sin^2 y \) is proved to be correct.

Key Concepts

Sum-to-Product Identities

Trigonometric identities are a fundamental part of understanding and solving problems in trigonometry. The sum-to-product identities help rewrite the sum or difference of two trigonometric terms into a product, making complex expressions simpler to handle.
To solve the exercise, you specifically use the sum-to-product identities for cosine:

• \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)
• \( \cos(a - b) = \cos a \cos b + \sin a \sin b \)

In the original equation, you expand \( \cos(x + y) \) and \( \cos(x - y) \) using these identities. These formulas break down complex angle expressions into simpler components that can be easily managed. This step transforms your problem into a manageable format, preparing it for further operations like multiplication.

Pythagorean Identity

The Pythagorean Identity is a vital tool in proving trigonometric identities. This identity states that:

• \( \cos^2 a + \sin^2 a = 1 \)

It establishes a fundamental relationship between the sine and cosine of an angle, allowing one to be rewritten in terms of the other.
During the solution, this identity was used to convert expressions involving \( \sin^2 x \) and \( \sin^2 y \) by expressing each as \( 1 - \cos^2 x \) and \( 1 - \cos^2 y \) respectively.
This pivotal manipulation simplifies the expression and ensures that the terms align with those on the right-hand side of the original equation. By substituting \( \cos^2 y \) with \( 1 - \sin^2 y \), you bridge the gap between unrelated terms, leading to a coherent and simplified expression.

Proving Trigonometric Identities

Proving trigonometric identities involves showing that two expressions are equivalent through a series of logical steps. This process requires a solid grasp of various trigonometric formulas and identities such as the Sum-to-Product Identities and the Pythagorean Identity.
In this case, the goal was to prove that \( \cos(x + y) \cos(x - y) = \cos^2 x - \sin^2 y \). The steps included:

• Expanding both sides using well-established identities
• Applying algebraic simplification techniques, such as leveraging the identity \((a-b)(a+b) = a^2 - b^2\)
• Using trigonometric identities to substitute and eliminate unnecessary terms
• Simplifying the equation back to the original statement

Through these structured steps, you transition from the original complex form to a clear and validated identity. This practice not only proves the statement but also deepens the understanding of how trigonometric identities interconnect.