Problem 70 - Precalculus with Limits

Question

In Exercises 61 - 70, prove the identity. \( \cos(x + y) + \cos(x - y) = 2 \cos x \cos y \)

Step-by-Step Solution

Answer

The given expression \( \cos(x + y) + \cos(x - y) \) is equal to \( 2 \cos x \cos y \), once the cosine of sum and difference identities are applied and simplified.

Step 1: Apply the Cosine of Sum Identity

Replace \( \cos(x + y) \) with its equivalent form using the cosine of sum identity. Thus, the expression becomes \( \cos(x + y) = \cos x \cos y - \sin x \sin y \).

Step 2: Apply the Cosine of Difference Identity

Replace \( \cos(x - y) \) with its equivalent form using the cosine of difference identity. The expression then becomes \( \cos(x - y) = \cos x \cos y + \sin x \sin y \).

Step 3: Combining the Two Equations

When adding the two expressions from Step 1 and Step 2, terms involving \( \sin x \sin y \) will cancel each other out, resulting in the final identity: \( 2 \cos x \cos y \).

Key Concepts

Cosine of Sum Identity

The cosine of sum identity is a cornerstone of trigonometry, which expresses the cosine of the sum of two angles in terms of the cosines and sines of those individual angles. Specifically, the formula is:
\[\begin{equation} \cos(x + y) = \cos x \cos y - \sin x \sin y \end{equation}\]
This identity allows us to transform the cosine of a combined angle into an expression involving the product of cosines and sines of the separate angles. By applying this identity, complex trigonometric expressions can often be simplified, which is especially handy for proving identities, solving equations, and integrating trigonometric functions. A fundamental skill in precalculus is recognizing when and how to effectively use this identity to maneuver between angles and their trigonometric functions.

Cosine of Difference Identity

Closely related to the cosine of sum identity is the cosine of difference identity, which follows a similar format but with a key difference in its sign. It is given by:
\[\begin{equation} \cos(x - y) = \cos x \cos y + \sin x \sin y \end{equation}\]
Comparing it to the cosine of sum identity, we see that while the cosine terms remain multiplied, the sine terms are now positively rather than negatively combined. This identity is invaluable when dealing with trigonometric functions involving the difference of angles. Understanding this identity can simplify the process of computing trigonometric values, prove other identities, or solve trigonometric equations, and is a vital part of a student's toolkit in precalculus.

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is fundamentally connected to geometric principles but goes deeper to explore the properties and applications of trigonometric functions such as sine, cosine, and tangent.
Trigonometry is not only about solving for angles and sides of triangles; it extends into modeling and solving real-world problems involving periodic phenomena, waves, and circular motion. When working on trigonometric identities, understanding the basic trigonometric functions and how to manipulate them using identities is crucial. This field forms the basis for more advanced studies in mathematics, physics, engineering, and even computer science.

Precalculus

Precalculus serves as the bridge between algebra and calculus, combining the skills and concepts from both to prepare students for the study of limits, derivatives, and integrals in calculus. Precalculus covers a variety of topics, including trigonometry, complex numbers, exponential and logarithmic functions, sequences, and series.
A strong foundation in precalculus is essential for students moving into calculus because it ensures that they have the algebraic and trigonometric skills necessary to grasp the more complex and abstract concepts they will encounter. In particular, a deep understanding of trigonometric identities, such as the cosine of sum and difference identities, allows students to simplify expressions and solve equations that are critical to calculus and other areas of advanced mathematics.