Problem 92 - Precalculus

Question

For the following exercises, prove the identities provided. $$ \cos (x+y) \cos (x-y)=\cos ^{2} x-\sin ^{2} y $$

Step-by-Step Solution

Answer

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

Step 1: Apply Sum-to-Product Identity

The sum-to-product identities relate the product of cosines to a sum of cosines. Use the identity \( \cos A \cos B = \frac{1}{2}(\cos(A+B) + \cos(A-B)) \). In this case, set \( A = x+y \) and \( B = x-y \). Then:\[ \cos(x+y)\cos(x-y) = \frac{1}{2}(\cos((x+y) + (x-y)) + \cos((x+y) - (x-y))) \] This simplifies to:\[ \cos(x+y)\cos(x-y) = \frac{1}{2}(\cos(2x) + \cos(2y)) \]

Step 2: Expand and Simplify

Now, expand and simplify the expression from Step 1:Notice that \( \cos(2x) = 2\cos^2(x) - 1 \) and \( \cos(2y) = 1 - 2\sin^2(y) \). Substitute these identities into the equation:\[ \cos(x+y)\cos(x-y) = \frac{1}{2}((2\cos^2(x) - 1) + (1 - 2\sin^2(y))) \]

Step 3: Simplifying Further

Combine like terms in the equation derived in the previous step:\[ \cos(x+y)\cos(x-y) = \frac{1}{2}(2\cos^2(x) - 2\sin^2(y)) \]Factor out the 2:\[ \cos(x+y)\cos(x-y) = \cos^2(x) - \sin^2(y) \]

Step 4: Final Step: Verification of Identity

Check if the simplified form matches the original identity to be proven. The final expression derived is \( \cos^2(x) - \sin^2(y) \), which matches the right-hand side of the given identity. Thus, the identity is verified.

Key Concepts

Sum-to-Product Identities

The sum-to-product identities are powerful tools in trigonometry that help simplify expressions involving the product of sine and cosine functions. These identities transform the product of trigonometric functions into sums or differences of trigonometric functions. For example, the identity \( \cos A \cos B = \frac{1}{2}(\cos(A+B) + \cos(A-B)) \) is part of this family.These identities are especially helpful when you encounter expressions with products of trigonometric functions because they transform them into more manageable sums. This is useful for simplification, integration, or even solving trigonometric equations.In the given exercise, using the sum-to-product identity allowed for the transformation of \( \cos(x+y)\cos(x-y) \) into a form involving the sums \( \cos(2x) + \cos(2y) \). This simplification is the first step in proving or verifying the identity provided in the original problem.

Cosine Function

The cosine function, often denoted as \( \cos \), is one of the fundamental trigonometric functions. It forms a relationship between the angle in a right triangle and the lengths of the adjacent side and hypotenuse. In a unit circle, the cosine of an angle \( x \) corresponds to the x-coordinate of a point on the circle.Key properties of the cosine function include:

• Periodic with a period of \( 2\pi \).
• Symmetric around the origin, making it an even function, i.e., \( \cos(-x) = \cos(x) \).
• Maximum and minimum values are 1 and -1, respectively.

Understanding these properties helps in working with identities involving cosine, such as the double angle identities used in the solution. By recognizing that \( \cos(2x) = 2\cos^2(x) - 1 \), we can expand and simplify expressions effectively, paving the way to verify complex trigonometric identities.

Double Angle Identities

Double angle identities are specific trigonometric identities that involve functions of angles double those in the original expressions. They are derived from basic trigonometric identities and are crucial for simplifying expressions and solving equations where angles are doubled.Common double angle identities include:

• \( \cos(2x) = 2\cos^2(x) - 1 \)
• \( \sin(2x) = 2\sin(x)\cos(x) \)
• \( \tan(2x) = \frac{2\tan(x)}{1-\tan^2(x)} \) \( (\tan(x) eq \pm 1) \)

In the solved exercise, the double angle identity \( \cos(2x) = 2\cos^2(x) - 1 \) was crucial in the simplification process. By substituting this identity in the expression, it allowed for reduction to simpler forms, leading to the solution: \( \cos^2(x) - \sin^2(y) \). Understanding these identities enhances one's ability to manipulate trigonometric functions efficiently, enabling proofs and solutions to complex trigonometric problems.