Problem 60 - Contemporary Precalculus

Question

$$\text { Prove the identity.}$$ $$\sin x \cos y=\frac{1}{2}[\sin (x+y)+\sin (x-y)]$$

Step-by-Step Solution

Answer

Question: Prove that $$\sin x \cos y = \frac{1}{2}[\sin (x+y)+\sin (x-y)]$$ using the sum and difference identities for sine functions. Solution Summary: To prove this identity, we first recall the sum and difference identities for the sine function. Next, we add these two identities together and simplify the expression. Finally, we divide both sides by 2 to obtain the given identity.

Step 1: Recall the sum and difference identities for sine function

The sum and difference identities for the sine function are given by: $$\sin(x + y) = \sin x \cos y + \cos x \sin y$$ $$\sin(x - y) = \sin x \cos y - \cos x \sin y$$

Step 2: Add the two identities from Step 1

By adding the two identities in Step 1, we get: $$\sin(x + y) + \sin(x - y) = (\sin x \cos y + \cos x \sin y) + (\sin x \cos y - \cos x \sin y)$$

Step 3: Simplify the expression obtained in Step 2

Simplify the right-hand side of the equation by combining like terms: $$\sin(x + y) + \sin(x - y) = 2 \sin x \cos y$$

Step 4: Divide both sides by 2

Divide both sides of the equation by 2 to isolate the term \(\sin x \cos y\): $$\frac{1}{2} (\sin(x + y) + \sin(x - y)) = \sin x \cos y$$ Now, we have successfully proven the given identity: $$\sin x \cos y = \frac{1}{2}[\sin (x+y)+\sin (x-y)]$$

Key Concepts

Sum and Difference Identities

When proving trigonometric identities, sum and difference identities are fundamental tools that come in handy. These identities show how to express the sine or cosine of a sum or difference of angles in terms of the sines and cosines of the individual angles.

Sum and difference identities for the sine function are particularly important for breaking down complex expressions into simpler components. For instance, the sum identity for sine is given by the equation \[ \sin(x + y) = \sin x \cos y + \cos x \sin y \],and conversely, the difference identity is written as\[ \sin(x - y) = \sin x \cos y - \cos x \sin y \].

Application of Sum and Difference Identities

By using these identities, you can transform a single sine term that involves a sum or difference into a more elaborate expression consisting of multiple sine and cosine terms. This expansion is particularly useful for proving other trigonometric identities, as it allows us to equate and manipulate different parts of an equation.

Sine Function Properties

The sine function has several intrinsic properties that are essential when proving trigonometric identities. Understanding these properties can greatly help simplify and solve problems involving sine.

Firstly, the sine function is periodic with a period of \(2\pi\), meaning that \(\sin(x + 2\pi) = \sin(x)\). This periodicity implies that the sine function repeats its values at regular intervals, which can be useful when solving trigonometric equations.

Another key property is that the sine function is odd, meaning that \(\sin(-x) = -\sin(x)\). This property becomes particularly useful when dealing with negative angles or when proving identities that involve subtraction.

Maximizing the Use of Sine Properties

Knowing these properties allows you to predict the behavior of the sine function under various transformations, including shifts and reflections, which can be critical when attempting to prove certain identities.

Trigonometric Identity Proof

The proof of trigonometric identities involves showing that two different trigonometric expressions are equivalent for all values of the variables they contain. The process usually entails algebraic manipulation, applying known identities, and adhering to the properties of trigonometric functions.

A methodical approach is important for clear and successful proofs. For the identity in the exercise given, the process includes recalling the appropriate sum and difference identities, strategically adding them, and simplifying to show the equivalence of the two sides of the equation.

Logical Flow in Identity Proofs

Ensuring a logical flow of steps is critical in trigonometric identity proofs. Each transformation must maintain the equation's balance, eventually arriving at a point where one side of the equation neatly transforms into the other, affirming their identity.

Mathematical Simplification

Mathematical simplification is the process of making a mathematical expression as simple as possible without changing its value. It is a crucial step in proving trigonometric identities and in solving equations in general. Common techniques include combining like terms, factoring, expanding, and canceling out terms.

In the context of trigonometric identities, simplification often involves adding or subtracting the identities, factoring out common terms, and dividing by constants to isolate the desired expression. An example of this is seen in the solution for the given exercise. By combining like terms and then dividing both sides of the equation by 2, we simplify the expression to prove the identity.

Mastering Simplification Techniques

Proficiency in simplification makes solving complex problems less daunting. Recognizing patterns and common terms can often lead the way to a more straightforward and elegant solution.