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.
(sec θ - 1) (sec θ + 1)
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.64
Textbook Question
Verify that each equation is an identity.
(csc θ + cot θ)/(tan θ + sin θ) = cot θ csc θ
Verified step by step guidance1
Step 1: Start by expressing all trigonometric functions in terms of sine and cosine. Recall that \( \csc \theta = \frac{1}{\sin \theta} \), \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
Step 2: Substitute these expressions into the left-hand side of the equation: \( \frac{\csc \theta + \cot \theta}{\tan \theta + \sin \theta} = \frac{\frac{1}{\sin \theta} + \frac{\cos \theta}{\sin \theta}}{\frac{\sin \theta}{\cos \theta} + \sin \theta} \).
Step 3: Simplify the numerator: \( \frac{1 + \cos \theta}{\sin \theta} \). Simplify the denominator: \( \frac{\sin^2 \theta + \sin \theta \cos \theta}{\cos \theta} \).
Step 4: Simplify the entire fraction by multiplying the numerator and the denominator by \( \cos \theta \) to eliminate the complex fraction: \( \frac{(1 + \cos \theta) \cos \theta}{\sin \theta (\sin \theta + \cos \theta)} \).
Step 5: Recognize that the expression simplifies to \( \cot \theta \csc \theta \) by using trigonometric identities and simplification techniques, thus 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 simplifying trigonometric expressions and verifying equations.
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Reciprocal Functions
Reciprocal functions in trigonometry include cosecant (csc), secant (sec), and cotangent (cot), which are defined as the reciprocals of sine, cosine, and tangent, respectively. For example, csc θ = 1/sin θ and cot θ = 1/tan θ. Recognizing these relationships is essential for manipulating and simplifying trigonometric expressions.
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Simplifying Trigonometric Expressions
Simplifying trigonometric expressions involves using identities and algebraic techniques to rewrite expressions in a more manageable form. This process often includes combining fractions, factoring, and substituting equivalent trigonometric functions. Mastery of simplification techniques is vital for verifying identities and solving trigonometric equations.
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