Problem 34 - Algebra and Trigonometry

Question

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 the Angle Sum and Difference Formulas

To prove the given identity, we start by applying the cosine angle sum and difference formulas. According to the angle sum formula, \( \cos(x + y) = \cos x \cos y - \sin x \sin y \).Similarly, for the angle difference formula, \( \cos(x - y) = \cos x \cos y + \sin x \sin y \).

Step 2: Add the Two Equations

Next, we add the two equations obtained from Step 1:\[ \cos(x + y) + \cos(x - y) = (\cos x \cos y - \sin x \sin y) + (\cos x \cos y + \sin x \sin y) \]

Step 3: Simplify the Expression

Combine the terms of the expression:\[ (\cos x \cos y - \sin x \sin y) + (\cos x \cos y + \sin x \sin y) = 2\cos x \cos y \]The terms \(-\sin x \sin y\) and \(+\sin x \sin y\) cancel each other out.

Step 4: Verify the Identity

The left side after simplification becomes:\[ 2 \cos x \cos y \]This matches the right side of the provided identity, thus verifying that \( \cos(x+y) + \cos(x-y) = 2 \cos x \cos y \) is true.

Key Concepts

Angle Sum Formula

The Angle Sum Formula is a critical tool in trigonometry that helps us calculate the sine or cosine of a sum or difference of angles. For the cosine function, the formula is expressed as \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). This is handy when you need to break down complicated trigonometric expressions into simpler parts.

Similarly, the angle difference formula is \( \cos(a - b) = \cos a \cos b + \sin a \sin b \). These formulas are foundational because they allow us to express complex trigonometric identities in terms of simpler ones.

When working with these formulas, remember:

• The role of angles can be swapped; the formula holds true because of the commutative property of addition.
• The difference in the sign between the sum and difference formulas is crucial and stems from the geometry of a unit circle.

Mastering these formulas will significantly enhance your capability to solve various trigonometric identities and problems effectively.

Cosine Function

The cosine function is one of the primary trigonometric functions, often represented as \( \cos \theta \), where \( \theta \) is the angle. It defines the relationship between the sides of a right triangle and its angles, as well as the horizontal coordinate of a point on the unit circle.

Key characteristics of the cosine function:

• Its range is between -1 and 1, inclusive.
• It is an even function, i.e., \( \cos(-\theta) = \cos(\theta) \).
• The function is periodic with a period of \( 2\pi \), meaning the pattern repeats every \( 2\pi \) units.

The cosine function forms the basis for many trigonometric identities, such as the one in our exercise, demonstrating its versatility in mathematical proof and calculation.

Mathematical Proof

A mathematical proof begins with assumptions, uses logical reasoning, and arrives at a conclusion that confirms a hypothesis. In trigonometry, proofs help establish the validity of identities like \( \cos(x+y) + \cos(x-y) = 2 \cos x \cos y \).

When constructing a proof, the following steps are important:

• Start with known identities or formulas, such as the angle sum and difference formulas in our exercise.
• Perform algebraic manipulations aimed at simplifying the expression or arriving at the desired form.
• Ensure each step follows logically from the previous one, eliminating or substituting terms as necessary.

Verifying an identity often requires checking both sides of the equation to confirm they are equivalent. Proofs solidify understanding of mathematical principles and reinforce problem-solving skills.