Textbook Question
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
csc θ - sin θ
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.70
Textbook Question
Verify that each equation is an identity.
(cot² t - 1)/(1 + cot² t) = 1 - 2 sin² t
Verified step by step guidance1
Step 1: Recall the Pythagorean identity: \(1 + \cot^2 t = \csc^2 t\).
Step 2: Substitute \(\csc^2 t\) for \(1 + \cot^2 t\) in the left-hand side of the equation: \((\cot^2 t - 1)/\csc^2 t\).
Step 3: Use the identity \(\csc^2 t = 1/\sin^2 t\) to rewrite the expression: \((\cot^2 t - 1) \cdot \sin^2 t\).
Step 4: Recall that \(\cot^2 t = \cos^2 t/\sin^2 t\), and substitute it into the expression: \((\cos^2 t/\sin^2 t - 1) \cdot \sin^2 t\).
Step 5: Simplify the expression to show that it equals \(1 - 2\sin^2 t\), verifying the identity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is crucial for verifying equations and simplifying expressions in trigonometry.
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Cotangent Function
The cotangent function, denoted as cot(t), is the reciprocal of the tangent function, defined as cot(t) = cos(t)/sin(t). It can also be expressed in terms of sine and cosine, which is essential for manipulating and simplifying trigonometric expressions. Recognizing how cotangent relates to other trigonometric functions is key to solving the given equation.
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Sine Function and Its Relationship to Cotangent
The sine function, sin(t), is a fundamental trigonometric function that relates to the cotangent function through the identity cot(t) = cos(t)/sin(t). The expression 1 - 2sin²(t) can be derived from the double angle formulas and is often used in conjunction with cotangent to verify identities. Understanding this relationship helps in transforming and equating both sides of the given equation.
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