Ch. 3 - Trigonometric Identities and Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problem 14

Chapter 3, Problem 14

Verify each identity. cos θ sec θ/cot θ= tan θ

Verified step by step guidance

Start by writing down the given expression clearly: \(\frac{\cos \theta \cdot \sec \theta}{\cot \theta} = \tan \theta\).

Recall the fundamental trigonometric identities: \(\sec \theta = \frac{1}{\cos \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).

Substitute these identities into the expression to rewrite it in terms of sine and cosine: \(\frac{\cos \theta \cdot \frac{1}{\cos \theta}}{\frac{\cos \theta}{\sin \theta}}\).

Simplify the numerator first: \(\cos \theta \cdot \frac{1}{\cos \theta} = 1\), so the expression becomes \(\frac{1}{\frac{\cos \theta}{\sin \theta}}\).

Simplify the complex fraction by multiplying numerator and denominator: \(\frac{1}{\frac{\cos \theta}{\sin \theta}} = \frac{\sin \theta}{\cos \theta}\), which is exactly \(\tan \theta\). This verifies 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 involving trigonometric functions that hold true for all values within their domains. Verifying an identity means showing that both sides simplify to the same expression using known identities, such as reciprocal, quotient, or Pythagorean identities.

Reciprocal and Quotient Identities

Reciprocal identities relate functions like sec θ = 1/cos θ and cot θ = 1/tan θ. Quotient identities express tan θ as sin θ/cos θ and cot θ as cos θ/sin θ. These relationships help rewrite expressions to simplify and verify identities.

Simplification Techniques in Trigonometry

Simplification involves rewriting expressions using algebraic manipulation and trigonometric identities to reduce complex fractions or products. Recognizing common factors and converting all terms to sine and cosine often makes verification straightforward.

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Fundamental Trigonometric Identities

Related Practice

Textbook Question

Use the given information to find the exact value of each of the following: sin 2θ

sin θ = ﹣2/3, θ lies in quadrant III.

Textbook Question

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. sin(60° - 45°)

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Textbook Question

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value. cos 4x + cos 2x

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Textbook Question

Use the given information to find the exact value of each of the following: cos 2θ

sin θ = ﹣2/3, θ lies in quadrant III.

Textbook Question

Find all solutions of each equation. tan x = 1

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Textbook Question

Use a sum or difference formula to find the exact value of each expression. cos(45° + 30°)