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Chapter 5: Problem 7

Simplify each of the following trigonometric expressions. $$ (\sin x-\cos x)(\sin x+\cos x) $$

Short Answer

Expert verified
The simplified expression is \(1 - 2(\cos x)^2\).

Step by step solution

01

Identify the Expression

The given expression is \((\sin x - \cos x)(\sin x + \cos x)\). It is in the form \((a-b)(a+b)\), which is a difference of squares.
02

Apply the Difference of Squares Formula

Recall that \((a-b)(a+b) = a^2 - b^2\). In this case, set \(a = \sin x\) and \(b = \cos x\). Substituting these into the formula: \((\sin x)^2 - (\cos x)^2\).
03

Simplify the Expression

The expression \((\sin x)^2 - (\cos x)^2\) can be further simplified. Use the Pythagorean identity \((\sin x)^2 + (\cos x)^2 = 1\). Rearrange it to express \((\sin x)^2\) as \(1 - (\cos x)^2\). The simplified form becomes \((1 - (\cos x)^2) - (\cos x)^2\).
04

Final Simplification

Simplify \(1 - 2(\cos x)^2\) to finalize the expression. This is the final simplified form of the original trigonometric expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares
The expression \((\sin x - \cos x)(\sin x + \cos x)\) is a perfect candidate for the difference of squares formula. The difference of squares is a handy algebraic tool that lets you simplify products that fit the pattern \((a-b)(a+b)\). When two quantities that differ only by the sign between them are multiplied, the formula \((a-b)(a+b) = a^2 - b^2\) is applicable. In this exercise:
  • Set \(a = \sin x\)
  • Set \(b = \cos x\)
Applying the difference of squares formula, the expression simplifies to \( (\sin x)^2 - (\cos x)^2 \), breaking it down easily into a more manageable equation.
Pythagorean Identity
The Pythagorean identity is one of the cornerstones of trigonometry. This identity states that:\[(\sin x)^2 + (\cos x)^2 = 1\]It's similar to the Pythagorean theorem in geometry and showcases the intrinsic relationship between sine and cosine functions.
Within the context of our expression, once we have \((\sin x)^2 - (\cos x)^2\), you can utilize this identity to further simplify it. By rearranging the Pythagorean identity, you can express \((\sin x)^2\) as \(1 - (\cos x)^2\).
Substituting this back into our expression gives us:\[(1 - (\cos x)^2) - (\cos x)^2\]This strategic move lets us simplify even further.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions is about reducing them to their simplest or most usable form. In our current work:We converted \((\sin x)^2 - (\cos x)^2\) into \(1 - 2(\cos x)^2\) using both the difference of squares and the Pythagorean identity.
This form, \(1 - 2(\cos x)^2\), is far simpler and more straightforward than what we started with.
By systematically applying known identities and algebraic techniques, complex expressions become not only more concise but also easier to interpret and work with in future calculations or problem-solving scenarios.

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Most popular questions from this chapter

Simplify each of the following trigonometric expressions. $$ \frac{\cot ^{2} x+1}{\csc x}-\csc x $$

Verify each of the trigonometric identities. $$ \frac{1}{1-\sin x}+\frac{1}{1+\sin x}=2 \sec ^{2} x $$

Simplify each expression using half-angle identities. Do not evaluate. $$ \sqrt{\frac{1-\cos \left(\frac{\pi}{4}\right)}{2}} $$

Verify the identities. $$ 2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)=\sin x $$

In Exercises 31-44, verify the identities. $$ \sin ^{2}\left(\frac{x}{2}\right)+\cos ^{2}\left(\frac{x}{2}\right)=1 $$
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