Problem 32 - Algebra and Trigonometry

Question

\(23-40\) . Prove the identity. $$ \cos (x+y)+\cos (x-y)=2 \cos x \cos y $$

Step-by-Step Solution

Answer

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

Step 1: Apply Cosine Addition Formulas

To solve the identity \( \cos(x+y) + \cos(x-y) = 2 \cos x \cos y \), we start by applying the cosine addition and subtraction formulas: - The formula for \( \cos(x+y) \) is \( \cos x \cos y - \sin x \sin y \). - The formula for \( \cos(x-y) \) is \( \cos x \cos y + \sin x \sin y \). Substitute these formulas into the left-hand side of the identity.

Step 2: Substitute and Simplify

Replace \( \cos(x+y) \) with \( \cos x \cos y - \sin x \sin y \) and \( \cos(x-y) \) with \( \cos x \cos y + \sin x \sin y \):\[ \cos(x+y) + \cos(x-y) = (\cos x \cos y - \sin x \sin y) + (\cos x \cos y + \sin x \sin y) \]Simplifying, we get:\[ \cos(x+y) + \cos(x-y) = 2\cos x \cos y \]Observe that \( -\sin x \sin y \) and \( +\sin x \sin y \) cancel each other out.

Step 3: Verify the Identity

After simplification, we have shown that the expression \( \cos(x+y) + \cos(x-y) \) simplifies directly to \( 2\cos x \cos y \). Therefore, the equation \( \cos(x+y) + \cos(x-y) = 2 \cos x \cos y \) holds true. This verifies the identity.

Key Concepts

Cosine Addition Formula

The cosine addition formula is a fundamental trigonometric identity that helps to express the cosine of the sum of two angles. It is expressed as: \( \cos(x+y) = \cos x \cos y - \sin x \sin y \).

This formula is crucial because it breaks down the cosine of an angle sum into products of sines and cosines of the individual angles, making calculations easier. Imagine you do not know the cosine of a complex angle; by breaking it into simpler components using this formula, solving the problem becomes straightforward.

It's especially useful in various fields, such as physics, engineering, and computer graphics, where complex angle manipulations are common. Whenever you encounter a complex angle, remember this formula to simplify your work.

Cosine Subtraction Formula

The cosine subtraction formula is closely related to the addition formula. It provides a way to express the cosine of the difference between two angles in terms of the cosines and sines of the individual angles. The formula is given by: \( \cos(x-y) = \cos x \cos y + \sin x \sin y \).

Here, the plus sign in front of \( \sin x \sin y \) signifies the operation of subtraction within the cosine function. This formula complements the cosine addition formula and is equally useful for breaking down complex angles into simpler, manageable terms.

In essence, the subtraction formula has real practical applications, such as solving trigonometric equations, analyzing wave functions, and other applications where phase differences and angular shifts are involved. Always rely on this formula to clearly understand trigonometric relationships.

Verify Trigonometric Identities

Verifying trigonometric identities is like solving a puzzle where each piece is connected to reveal a clear and consistent picture. This process involves proving that both sides of an equation are equal to each other by using known trigonometric formulas and identities. For our exercise, simplifying \( \cos(x+y) + \cos(x-y) \) using the addition and subtraction formulas leads directly to \( 2\cos x \cos y \).

To verify an identity, follow these steps:

• Break down complex expressions into simpler trigonometric functions using known identities like addition or subtraction formulas.
• Simplify both sides if necessary to see if they match.
• Watch for cancellations, such as terms that negate each other, which streamline expressions.

This verification process not only refines your problem-solving skills but also deepens your understanding of how trigonometric functions interact. It's essential in advanced mathematics, helping validate solutions in calculus, engineering, and physics. Practice consistently, and you'll master these techniques in no time.