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Chapter 5: Problem 52

Prove that each of the following identities is true: $$ \cos ^{4} A-\sin ^{4} A=1-2 \sin ^{2} A $$

Short Answer

Expert verified
The identity is true because \( \cos^4 A - \sin^4 A \) simplifies to \( 1 - 2\sin^2 A \).

Step by step solution

01

Apply the Difference of Squares

The expression \( \cos^4 A - \sin^4 A \) matches the pattern of a difference of squares, which is given by \( a^2 - b^2 = (a-b)(a+b) \). Here, set \( a = \cos^2 A \) and \( b = \sin^2 A \). Thus, we can rewrite \( \cos^4 A - \sin^4 A \) as \((\cos^2 A - \sin^2 A)(\cos^2 A + \sin^2 A)\).
02

Simplify Using Pythagorean Identity

The Pythagorean identity states that \( \cos^2 A + \sin^2 A = 1 \). Substituting this into the expression from Step 1, we have \( (\cos^2 A - \sin^2 A) \times 1 = \cos^2 A - \sin^2 A \). So, \( \cos^4 A - \sin^4 A = \cos^2 A - \sin^2 A \).
03

Convert \( \cos^2 A - \sin^2 A \) to Trigonometric Identity

We know from trigonometric identities that \( \cos^2 A - \sin^2 A \) can be expressed as \( 1 - 2\sin^2 A \). This conversion is based on the identity \( \cos^2 A = 1 - \sin^2 A \), thus \( \cos^2 A - \sin^2 A = (1 - \sin^2 A) - \sin^2 A = 1 - 2\sin^2 A \).
04

Verify the Equality

At this point, we have shown that \( \cos^4 A - \sin^4 A \) simplifies to \( 1 - 2\sin^2 A \) via trigonometric identities. Therefore, the original identity \( \cos^4 A - \sin^4 A = 1 - 2\sin^2 A \) is indeed true.

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Key Concepts

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

Difference of Squares
The difference of squares is a handy algebraic tool that allows you to break down expressions into simpler components. When you have two square terms subtracted from each other, like in
  • \( a^2 - b^2 \)
This can be rewritten as
  • \((a - b)(a + b)\)
In the problem we encountered, the expression \( \cos^4 A - \sin^4 A \) fits the difference of squares pattern. Here, you set
  • \( a = \cos^2 A \) and
  • \( b = \sin^2 A \)
This means we can rewrite the expression as:
  • \((\cos^2 A - \sin^2 A)(\cos^2 A + \sin^2 A)\)
Breaking down complex expressions using the difference of squares can simplify the steps required to solve trigonometric problems, making them far more approachable.
Pythagorean Identity
Pythagorean Identity is a fundamental building block in trigonometry. It states that
  • \( \cos^2 A + \sin^2 A = 1 \)
This relationship comes from the Pythagorean theorem when applied to the unit circle. In simpler terms, if you take any angle \( A \) and find its cosine and sine squared values, their sum will always equal one.
In the solution to the original exercise, this identity was used to simplify the expression
  • \((\cos^2 A - \sin^2 A)(\cos^2 A + \sin^2 A)\)
Because \( \cos^2 A + \sin^2 A = 1 \), the expression reduced to
  • \( \cos^2 A - \sin^2 A \)
Recognizing and applying the Pythagorean identity allows you to find simplicity amidst seemingly complex trigonometric equations.
Trigonometric Expressions
Trigonometric expressions like \( \cos^2 A - \sin^2 A \) often have alternative forms thanks to identities that relate different trigonometric functions. In this case, the expression can be reworked using the identity
  • \( \cos^2 A = 1 - \sin^2 A \)
In the given problem, it evolved to
  • \( \cos^2 A - \sin^2 A = 1 - 2\sin^2 A \)
This is a trigonometric identity that transforms our equation into familiar terms, making it easier to grasp and verify its truth.
Interchanging and expressing trigonometric expressions in different ways is essential for solving a broad range of trigonometric problems. By using these identities, you can simplify equations, making them much more manageable.

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Most popular questions from this chapter

Prove that each of the following identities is true: $$ \frac{1-\cos ^{4} \theta}{1+\cos ^{2} \theta}=\sin ^{2} \theta $$

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The problems that follow review material we covered in Section 4.3. Graph one complete cycle. $$ y=\cos \left(x-\frac{\pi}{3}\right) $$

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