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Magazine Chapter 7: Problem 45
Verify that each trigonometric equation is an identity. $$\sin ^{4} \theta-\cos ^{4} \theta=2 \sin ^{2} \theta-1$$
Short Answer
Expert verified
Both sides simplify to \- \cos (2 \theta ) \.
Step by step solution
01
- Simplify the Left Side Using Trigonometric Identities
Start by recognizing that the left side \(\sin ^{4} \theta-\cos ^{4} \theta\) can be factored using the difference of squares. This gives: \(\sin ^{4} \theta-\cos ^{4} \theta = (\sin^2 \theta - \cos^2 \theta)(\sin^2 \theta + \cos^2 \theta) \)
02
- Simplify Using Pythagorean Identity
Recall the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Replace the \(\sin^2 \theta + \cos^2 \theta\) in the equation: \( \left( \sin^2 \theta - \cos^2 \theta \right) ( 1 ) \) Since multiplying by 1 does not change the expression, it simplifies to: \( \sin^2 \theta - \cos^2 \theta \)
03
- Use the Double-Angle Identity
Rewrite \(\sin^2 \theta - \cos^2 \theta\) using the double-angle identity for cosine, \( \cos ( 2 \theta ) = \cos^2 \theta - \sin^2 \theta\). Thus, \(\sin^2 \theta - \cos^2 \theta = -( \cos (2 \theta ) )\).This means: \( \sin ^{4} \theta-\cos ^{4} \theta = -\cos (2 \theta ) \)
04
- Verify with Original Right Side
Finally, simplify the original right side \( 2 \sin^2 \theta - 1 \). Using the double angle identity for sine, \( \sin^2 \theta = \frac{1 - \cos (2 \theta)}{2} \), we have: \( 2 \sin^2 \theta - 1 = 2 \left ( \frac{1 - \cos (2 \theta)}{2} \right ) - 1 \) This simplifies to: \( 2 \left ( \frac{1 - \cos (2 \theta)}{2} \right ) - 1 = (1 - \cos (2 \theta)) - 1 = - \cos (2 \theta) \)
05
Conclusion - Verify Both Sides Are Equal
Both the left and right sides simplify to \- \cos (2 \theta )\. Thus, the given trigonometric equation \( \sin^{4} \theta - \cos^{4} \theta = 2 \sin^{2} \theta - 1 \) is verified to be an identity.
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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 powerful algebraic tool. It states that \(a^2 - b^2 = (a - b)(a + b)\). This identity is useful for factoring complex expressions. In this problem, we started with \(\sin^4 \theta - \cos^4 \theta\). By recognizing that \(\sin^4 \theta\) and \(\cos^4 \theta\) are both squares, we rewrote the expression as \(\sin^2 \theta \cdot \sin^2 \theta - \cos^2 \theta \cdot \cos^2 \theta\). Applying the difference of squares identity transformed it to \(\sin^2 \theta - \cos^2 \theta\) times \(\sin^2 \theta + \cos^2 \theta\). This factorization simplifies solving many trigonometric equations.
Pythagorean identity
The Pythagorean identity is fundamental in trigonometry. It states that \(\sin^2 \theta + \cos^2 \theta = 1\). This identity helps bridge different trigonometric functions. In this exercise, we leveraged the Pythagorean identity to simplify \(\sin^2 \theta + \cos^2 \theta\) in our expression. By substituting \(1\) for \(\sin^2 \theta + \cos^2 \theta\), we turned the complex product \(\sin^2 \theta - \cos^2 \theta \cdot (\sin^2 \theta + \cos^2 \theta)\) into just \(\sin^2 \theta - \cos^2 \theta\). This simplification is crucial in many trigonometric proofs and problems.
double-angle identity
The double-angle identities allow you to express trigonometric functions of double angles (like \(2\theta\)) in terms of single angles. For cosine, the double-angle identity is \(\cos(2\theta) = \cos^2 \theta - \sin^2 \theta\). By rearranging it, we get \(\sin^2 \theta - \cos^2 \theta = -\cos(2\theta)\). In our problem, this conversion was crucial to match both sides of the equation. Substituting \(-\cos(2\theta)\) for \(\sin^2 \theta - \cos^2 \theta\) simplified the left side to resemble the right side. Mastering double-angle identities is important for advanced trigonometry.
proof by simplification
Proof by simplification involves breaking down a complex expression to verify its equality. Here, we started with \(\sin^4 \theta - \cos^4 \theta\) and transformed it step by step using different identities. By employing the difference of squares, Pythagorean identity, and double-angle identity, we systematically simplified both sides. Eventually, we showed that the left side equaled the right side, confirming the original equation. This methodical simplification proves that \(\sin^4 \theta - \cos^4 \theta = 2 \sin^2 \theta - 1\) holds true. Using multiple methods to simplify both sides of an equation is key in trigonometric proofs.
