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Chapter 10: Problem 92

Verify that equation is an identity. \(\left(1-\cos ^{2} \alpha\right)\left(1+\cos ^{2} \alpha\right)=2 \sin ^{2} \alpha-\sin ^{4} \alpha\)

Short Answer

Expert verified
The equation is an identity.

Step by step solution

01

Expand the Left Side

Expand the expression on the left side of the equation: \[ (1 - \cos^2 \alpha)(1 + \cos^2 \alpha) \] Using the difference of squares formula: \[ a^2 - b^2 = (a + b)(a - b) \] Letting \(a = 1\) and \(b = \cos^2 \alpha\), we find: \[ 1^2 - (\cos^2 \alpha)^2 = 1 - \cos^4 \alpha \]
02

Transform Using Trigonometric Identity

Recognize that \(1 - \cos^2 \alpha = \sin^2 \alpha\) based on the Pythagorean identity. So, we continue: \[ 1 - \cos^4 \alpha = (1 - \cos^2 \alpha)(1 + \cos^2 \alpha) \] Substitute \(\sin^2 \alpha\) for \(1 - \cos^2 \alpha\): \[ 1 - \cos^4 \alpha = \sin^2 \alpha \cdot (1 + \cos^2 \alpha) \]
03

Expand Using the Inner Term

Expand \(\sin^2 \alpha \cdot (1 + \cos^2 \alpha)\):\[ \sin^2 \alpha \cdot 1 + \sin^2 \alpha \cdot \cos^2 \alpha = \sin^2 \alpha + \sin^2 \alpha \cos^2 \alpha \]
04

Simplify and Compare

Notice that \(\sin^4 \alpha = (\sin^2 \alpha)^2\), so:\[ \sin^2 \alpha + \sin^2 \alpha \cos^2 \alpha = \sin^2 \alpha(1 + \cos^2 \alpha) = 2 \sin^2 \alpha - (\sin^2 \alpha)^2 \]Hence:\[ 1 - \cos^4 \alpha \equiv 2 \sin^2 \alpha - \sin^4 \alpha \] This establishes that both sides are identical.

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Key Concepts

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

Difference of Squares
The concept of the difference of squares is a foundational algebraic identity. It states that the difference between the squares of two terms, \( a^2 - b^2 \), can be expressed as the product of a sum and a difference: \((a + b)(a - b)\). This identity is useful in simplifying expressions and solving equations.
  • In our problem, the expression \((1 - \cos^2 \alpha)(1 + \cos^2 \alpha)\) is an example of the difference of squares, where \(a = 1\) and \(b = \cos^2 \alpha\).
  • Applying the identity, we find \(1^2 - (\cos^2 \alpha)^2 = 1 - \cos^4 \alpha\).
  • This simplification is crucial because it enables us to transform and compare the expression more easily with the given identity.
Mastering the difference of squares allows you to handle more complex algebraic expressions efficiently. It's a neat trick for quickly factoring expressions.
Pythagorean Identity
The Pythagorean identity is a fundamental trigonometric identity which states \(\sin^2 \alpha + \cos^2 \alpha = 1\). This identity is pivotal in trigonometry because it connects sine and cosine, allowing one to be expressed in terms of the other.
  • In this problem, we use a rearranged form of the Pythagorean identity: \(1 - \cos^2 \alpha = \sin^2 \alpha\). This substitution is key to simplifying our expression.
  • By replacing \(1 - \cos^2 \alpha\) with \(\sin^2 \alpha\), we make it easier to manipulate and evaluate the trigonometric expressions involved.
  • Understanding and applying the Pythagorean identity can simplify many problems in trigonometry, making it easier to solve or verify complex equations.
The Pythagorean identity is a tool you will use time and again in trigonometry. It highlights the interconnected nature of trigonometric functions.
Expanding Expressions
Expanding expressions involves distributing terms within a parenthesis and combining like terms to simplify or transform equations. This technique is extremely common in algebra and calculus.
  • When we expand \(\sin^2 \alpha \cdot (1 + \cos^2 \alpha)\), we multiply each term in the parenthesis by \(\sin^2 \alpha\). The result is \( \sin^2 \alpha + \sin^2 \alpha \cos^2 \alpha\).
  • Further simplifying the expression involves recognizing that \(\sin^4 \alpha\) is \((\sin^2 \alpha)^2\), leading us to \(2 \sin^2 \alpha - \sin^4 \alpha\).
  • Breaking things down in steps, as we've done here, is crucial for comparing it to the original identity to verify that both sides of the equation are equivalent.
By expanding expressions adeptly, you can unlock the full potential of algebraic and trigonometric identities. It helps you bridge the gap between seemingly complex equations and their simplified forms.

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Most popular questions from this chapter

Verify that each equation is an identity. $$\frac{\cos (A-B)}{\sin (A+B)}=\frac{1+\cot A \cot B}{\cot A+\cot B}$$

Verify that each equation is an identity. $$\sin \left(210^{\circ}+x\right)-\cos \left(120^{\circ}+x\right)=0$$

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximale values in radians to four decimal places and approximate values in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\sin \theta-\sin 2 \theta=0$$

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximale values in radians to four decimal places and approximate values in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\cos x=\sin ^{2} \frac{x}{2}$$

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximale values in radians to four decimal places and approximate values in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\cos \frac{\theta}{2}=1$$
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