Problem 36
Question
In Exercises 29-36, use a double-angle formula to rewrite the expression. \( (\sin x - \cos x)(\sin x + \cos x) \)
Step-by-Step Solution
Verified Answer
The simplification of the expression \( (\sin x - \cos x)(\sin x + \cos x) \) per double-angle formulas is \( -\cos 2x \).
1Step 1: Identify the Expression
We have a trigonometric expression \( (\sin x - \cos x)(\sin x + \cos x) \). This expression should be recognized as a difference of squares.
2Step 2: Simplify the Expression
In the form of \( a^2 - b^2 \), the difference of squares simplifies to \( a - b \) times \( a + b \). In this exercise, \( a \) corresponds to \( \sin x \) and \( b \) corresponds to \( \cos x \). Thus, applying the formula, we get \( \sin^2 x - \cos^2 x \).
3Step 3: Apply the Trigonometric Identity
We can rewrite \( \sin^2 x - \cos^2 x \) using the double-angle formula. According to trigonometric identity, \( \sin^2 x - \cos^2 x = -\cos 2x \).
Key Concepts
Trigonometric IdentitiesDifference of SquaresTrigonometric Expressions
Trigonometric Identities
Understanding trigonometric identities can be a game-changer when it comes to simplifying complex trigonometric expressions. In essence, these identities are equations that hold true for any value inserted into the trigonometric functions.
One of the fundamental identities is the Pythagorean identity, which states that for any angle \( x \), \( \sin^2 x + \cos^2 x = 1 \). Other commonly used trigonometric identities include the double-angle formulas, which provide expressions for \( \sin 2x \), \( \cos 2x \), and \( \tan 2x \) in terms of \( \sin x \) and \( \cos x \).
For example, \( \cos 2x \) can be represented in three different ways: \( \cos^2 x - \sin^2 x \), \( 2\cos^2 x - 1 \), or \( 1 - 2\sin^2 x \). These identities are incredibly useful for exercises that involve rewriting expressions such as \( \sin x - \cos x \), since they allow us to simplify the terms into a more manageable form.
The power of trigonometric identities lies in their ability to transform expressions into equivalent forms that might offer a clearer path to simplification or solution. Whether it's a double-angle formula or any other identity, mastering these equations can significantly aid in solving a wide range of trigonometry problems.
One of the fundamental identities is the Pythagorean identity, which states that for any angle \( x \), \( \sin^2 x + \cos^2 x = 1 \). Other commonly used trigonometric identities include the double-angle formulas, which provide expressions for \( \sin 2x \), \( \cos 2x \), and \( \tan 2x \) in terms of \( \sin x \) and \( \cos x \).
For example, \( \cos 2x \) can be represented in three different ways: \( \cos^2 x - \sin^2 x \), \( 2\cos^2 x - 1 \), or \( 1 - 2\sin^2 x \). These identities are incredibly useful for exercises that involve rewriting expressions such as \( \sin x - \cos x \), since they allow us to simplify the terms into a more manageable form.
The power of trigonometric identities lies in their ability to transform expressions into equivalent forms that might offer a clearer path to simplification or solution. Whether it's a double-angle formula or any other identity, mastering these equations can significantly aid in solving a wide range of trigonometry problems.
Difference of Squares
The concept of the difference of squares is a fundamental idea in algebra that provides a simple way to factorize certain types of expressions. It is based on the observation that the product of the sum and the difference of two terms \( a + b \) and \( a - b \) results in the difference of their squares: \( a^2 - b^2 \).
The beauty of this algebraic pattern is its straightforward application, which can be extended to many mathematical scenarios, including trigonometric expressions. When we encounter a term like \( (\sin x - \cos x)(\sin x + \cos x) \), recognizing it as a difference of squares allows us to immediately simplify it to \( \sin^2 x - \cos^2 x \) without multiplying each term by its counterpart.
This simplification is particularly useful in trigonometry, where dealing with squared trigonometric functions can swiftly become complicated. By using the difference of squares, we can often find smoother paths to solutions, making complex problems more approachable.
The beauty of this algebraic pattern is its straightforward application, which can be extended to many mathematical scenarios, including trigonometric expressions. When we encounter a term like \( (\sin x - \cos x)(\sin x + \cos x) \), recognizing it as a difference of squares allows us to immediately simplify it to \( \sin^2 x - \cos^2 x \) without multiplying each term by its counterpart.
This simplification is particularly useful in trigonometry, where dealing with squared trigonometric functions can swiftly become complicated. By using the difference of squares, we can often find smoother paths to solutions, making complex problems more approachable.
Trigonometric Expressions
Trigonometric expressions are combinations of trigonometric functions, numbers, and variables that we can manipulate using algebraic operations and trigonometric identities. The ability to simplify and rewrite these expressions is crucial in solving trigonometry problems and proving identities.
For instance, in the exercise \( (\sin x - \cos x)(\sin x + \cos x) \), we start with an expression that involves both the sine and cosine functions. By recognizing it as a difference of squares—as discussed previously—we simplify it to \( \sin^2x - \cos^2x \).
From here, instead of being stuck with an expression that seems complex, we can apply a double-angle formula. In this case, we use the identity \( \sin^2 x - \cos^2 x = -\cos 2x \) to further simplify the expression. This process showcases how a comprehension of trigonometric expressions and their identities can lead to a deeper understanding and a quicker resolution of problems.
For instance, in the exercise \( (\sin x - \cos x)(\sin x + \cos x) \), we start with an expression that involves both the sine and cosine functions. By recognizing it as a difference of squares—as discussed previously—we simplify it to \( \sin^2x - \cos^2x \).
From here, instead of being stuck with an expression that seems complex, we can apply a double-angle formula. In this case, we use the identity \( \sin^2 x - \cos^2 x = -\cos 2x \) to further simplify the expression. This process showcases how a comprehension of trigonometric expressions and their identities can lead to a deeper understanding and a quicker resolution of problems.
Other exercises in this chapter
Problem 35
In Exercises 25-38, find all solutions of the equation in the interval \( [0, 2\pi) \). \( 2 \sec^2 x + \tan^2 x - 3 = 0 \)
View solution Problem 35
In Exercises 9-50, verify the identity \( \dfrac{\tan x \cot x}{\cos x} = \sec x \)
View solution Problem 36
In Exercises 29 - 36, write the expression as the sine, cosine,or tangent of an angle. \( \cos 3x \cos 2y + \sin 3x \sin 2y \)
View solution Problem 36
In Exercises 25-38, find all solutions of the equation in the interval \( [0, 2\pi) \). \( \cos x + \sin x \tan x = 2 \)
View solution