Problem 60 - Contemporary Precalculus

Question

Prove the given sum to product identity. $$\cos x+\cos y=2 \cos \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right)$$

Step-by-Step Solution

Answer

Question: Prove the sum-to-product identity: $$\cos x+\cos y=2 \cos \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right)$$ Answer: Using the angle addition formula for cosine and manipulating the expressions, we have successfully proved the given sum-to-product identity: $$\cos x+\cos y=2 \cos \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right)$$

Step 1: Initially Manipulate the RHS of the Identity

We must express the RHS of the given identity in terms of \(x\) and \(y\). Let's first rewrite the RHS in a more convenient form: $$2\cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right)$$

Step 2: Apply Angle Addition Formula to the RHS

Let's use the angle addition formula in the RHS expression so we can reach our goal: $$2\cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) = 2 \left[\cos\left(\frac{x+y}{2}+\frac{x-y}{2}\right) + \cos\left(\frac{x+y}{2}-\frac{x-y}{2}\right) \right]$$ We can simplify this expression: $$= 2 \left[\cos(x) + \cos(y) \right]$$

Step 3: Compare the LHS and RHS

Now, let's compare the initial given identity: $$\cos x+\cos y = 2 \cos \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right)$$ with our derived expression: $$\cos x+\cos y = 2 \left[\cos(x) + \cos(y) \right]$$ As both the LHS and RHS expressions are equal, we have successfully proved the given sum-to-product identity: $$\cos x+\cos y=2 \cos \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right)$$

Key Concepts

Sum-to-Product Identities

Sum-to-product identities are a key part of trigonometry, simplifying the process of manipulating trigonometric expressions. They convert the sum or difference of trigonometric functions into a product. This can make complex expressions more manageable.
Let's take a specific example: the identity \( \cos x + \cos y = 2 \cos \left( \frac{x+y}{2} \right) \cos \left( \frac{x-y}{2} \right) \). In this context, the identity converts a sum of two cosine terms into a product, simplifying calculations in many problems.
Using the sum-to-product identities helps in:

• Solving trigonometric equations.
• Simplifying integrals involving trigonometric functions.
• Transforming expressions in physics and engineering.

By expressing trigonometric functions as products, these identities enable easier manipulation and integration, which is especially useful in calculus and differential equations.

Trigonometric Proofs

Trigonometric proofs involve deriving one trigonometric equation from another, using identities and properties of trigonometric functions. They help strengthen your understanding of trig identities by showcasing logical reasoning and algebraic manipulation.
In our exercise, we have proved the identity \( \cos x + \cos y = 2 \cos \left( \frac{x+y}{2} \right) \cos \left( \frac{x-y}{2} \right) \). The key steps for any trigonometric proof involve:

• Analyzing the given identity to understand each side of the equation.
• Using known identities such as sum-to-product and angle addition to express one side of the equation in terms of the other.
• Simplifying the expressions systematically to arrive at a recognizable form, proving LHS = RHS.

Trigonometric proofs often require creativity and practice. They teach important skills like algebraic manipulation and attentiveness to detail, which can be broadly applicable in mathematics.

Angle Addition Formula

The angle addition formulas provide a way to express trigonometric functions of summed angles in terms of the trigonometric functions of the individual angles. For cosine, it is expressed as \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).
In proving the sum-to-product identity, we used a concept similar to angle addition formulas to rewrite and simplify expressions. Through substitution and strategic arrangement of angles like \( \frac{x+y}{2} \) and \( \frac{x-y}{2} \), you derive terms that match the identity.
Here's why angle addition formulas are so useful:

• They allow the transformation of complex expressions into simpler ones.
• Help derive many other trigonometric identities used in calculus.
• Are essential in solving equations and proving other theorems in trigonometry.

Understanding and applying the angle addition formulas can open doors to solving a wide array of problems in mathematics and physics, laying a solid foundation for more advanced studies.