Textbook Question
Match each expression in Column I with its equivalent expression in Column II.
sin 60° cos 45° - cos 60° sin 45°
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6. Trigonometric Identities and More Equations
Sum and Difference Identities
Problem 73
Textbook Question
Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos 2x = (cot² x - 1)/(cot² x + 1)
Verified step by step guidance1
Start by recalling the double-angle identity for cosine: \(\cos 2x = \cos(x + x) = \cos^2 x - \sin^2 x\).
Express \(\cot x\) in terms of sine and cosine: \(\cot x = \frac{\cos x}{\sin x}\), so \(\cot^2 x = \frac{\cos^2 x}{\sin^2 x}\).
Rewrite the right-hand side of the equation \(\frac{\cot^2 x - 1}{\cot^2 x + 1}\) by substituting \(\cot^2 x\) with \(\frac{\cos^2 x}{\sin^2 x}\), giving \(\frac{\frac{\cos^2 x}{\sin^2 x} - 1}{\frac{\cos^2 x}{\sin^2 x} + 1}\).
Simplify the complex fraction by multiplying numerator and denominator by \(\sin^2 x\) to eliminate the denominators inside the fraction, resulting in \(\frac{\cos^2 x - \sin^2 x}{\cos^2 x + \sin^2 x}\).
Use the Pythagorean identity \(\cos^2 x + \sin^2 x = 1\) to simplify the denominator, so the expression becomes \(\cos^2 x - \sin^2 x\), which matches the double-angle identity for \(\cos 2x\), verifying the identity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Double-Angle Identity for Cosine
The double-angle identity expresses cos 2x in terms of functions of x, such as cos 2x = cos² x - sin² x or cos 2x = 2 cos² x - 1. This identity helps rewrite trigonometric expressions involving 2x into simpler forms involving x, facilitating verification of equations.
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Double Angle Identities
Cotangent and Its Relationship to Sine and Cosine
Cotangent is defined as cot x = cos x / sin x. Understanding this relationship allows conversion of expressions involving cot² x into sine and cosine terms, which is essential for manipulating and simplifying the given equation.
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Trigonometric Identities and Algebraic Manipulation
Verifying identities requires applying known trigonometric identities and algebraic techniques such as factoring, common denominators, and substitution. This process transforms one side of the equation to match the other, confirming the identity's validity.
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