Problem 41 - Precalculus: Functions and Graphs

Question

Exer. \(37-46:\) Verify the identity. $$\cos (u+v)+\cos (u-v)=2 \cos u \cos v$$

Step-by-Step Solution

Answer

Identity verified: \( \cos (u+v)+\cos (u-v)=2 \cos u \cos v \).

Step 1: Recall the sum-to-product identities

To verify the given trigonometric identity, we need to recall the sum-to-product formulas. The relevant identity here is:\[ \cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \]

Step 2: Substitute specific expressions

In our problem, we are given \( A = u+v \) and \( B = u-v \). Substitute these into the sum-to-product identity:\[ \cos(u+v) + \cos(u-v) = 2 \cos \left( \frac{(u+v) + (u-v)}{2} \right) \cos \left( \frac{(u+v) - (u-v)}{2} \right) \]

Step 3: Simplify the expressions inside the cosines

Simplify the terms inside the arguments of the cosine functions:- For the first cosine function, compute \( \frac{(u+v) + (u-v)}{2} \): \[ \frac{2u}{2} = u \]- For the second cosine function, compute \( \frac{(u+v) - (u-v)}{2} \): \[ \frac{2v}{2} = v \]

Step 4: Conclude with simplified product identity

Substitute back the simplified terms into the identity:\[ \cos(u+v) + \cos(u-v) = 2 \cos(u) \cos(v) \]This shows that the original identity does indeed hold true.

Key Concepts

Sum-to-Product Identities

Trigonometric identities like the sum-to-product identities help convert sums or differences of trigonometric functions into products, simplifying complex expressions. These identities are especially useful for verifying identities or solving equations. The formula for the cosine sum-to-product identity is particularly useful for rewriting the original problem:

• \( \cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \)

In this instance, the identity simplifies handling trigonometric properties, making equations easier to solve or verify. Remember, using these identities helps when managing complex expressions by turning an addition problem into a multiplicative one.

Cosine Function

The cosine function, noted as \( \cos \theta \), is one of the primary functions in trigonometry and relates the angle to the adjacent side over the hypotenuse in a right-angled triangle. It's periodic, with a period of \( 2\pi \), meaning it repeats its values in this interval. This periodic nature allows it to be used extensively in wave calculations and in analyzing cycles.

• Key properties include evenness: \( \cos(-\theta) = \cos(\theta) \)
• Its range is between [-1, 1], offering predictable bounds which make cosine a stable function for various calculations

When undergoing transformations or verifying identities like in our problem, understanding these foundational aspects of the cosine function is crucial.

Trigonometric Proofs

Proofs in trigonometry often involve verifying an identity or transforming one expression into another equivalent one. In our exercise, we proved an identity using transformation methods.
Trigonometric proofs require:

• Understanding of basic trigonometric identities, like the Pythagorean identities
• A strategy for substituting expressions back and forth using known identities
• Logical steps that follow from expressing complex functions in simpler terms

By rewriting the cosine identity as a product, we were able to mathematically show that both sides of the equation balance, demonstrating the truth of the given identity.

Precalculus

Precalculus serves as a foundation for calculus, covering essential concepts like trigonometry, functions, and algebra that are integral for advanced mathematics. Understanding identities is a significant part of precalculus, as it helps students see the interrelations within mathematics.

Trigonometric identities not only ease computations but also provide insight into the coherence of trigonometric concepts. Learning how to apply these identities strengthens analytical skills and prepares students for challenges in calculus, where these concepts are expanded upon. By practicing trigonometric proofs and identities in precalculus, students build a robust base for more advanced studies.