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Ch. 5 - Trigonometric Identities
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 5 - Trigonometric IdentitiesProblem 50
Chapter 6, Problem 50

Verify that each equation is an identity.
tan (θ/2) = csc θ - cot θ

Verified step by step guidance
1
Start by recalling the half-angle identity for tangent: \(\tan\left(\frac{\theta}{2}\right) = \frac{\sin \theta}{1 + \cos \theta}\) or alternatively \(\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos \theta}{\sin \theta}\). Choose the form that seems easier to work with for this problem.
Rewrite the right-hand side expression \(\csc \theta - \cot \theta\) in terms of sine and cosine functions: \(\csc \theta = \frac{1}{\sin \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). So, \(\csc \theta - \cot \theta = \frac{1}{\sin \theta} - \frac{\cos \theta}{\sin \theta}\).
Combine the terms on the right-hand side over a common denominator \(\sin \theta\): \(\frac{1 - \cos \theta}{\sin \theta}\).
Compare this simplified right-hand side expression \(\frac{1 - \cos \theta}{\sin \theta}\) with the half-angle identity form of \(\tan\left(\frac{\theta}{2}\right)\) you selected in step 1. They should match, confirming the identity.
Conclude that since both sides simplify to the same expression, the given equation \(\tan\left(\frac{\theta}{2}\right) = \csc \theta - \cot \theta\) is indeed an 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 both sides simplify to the same expression, often by using fundamental identities like Pythagorean or reciprocal identities.
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Fundamental Trigonometric Identities

Half-Angle Formulas

Half-angle formulas express trigonometric functions of half an angle in terms of the full angle. For example, tan(θ/2) can be written using sine and cosine of θ, which helps in transforming and simplifying expressions involving half angles.
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Quadratic Formula

Reciprocal and Quotient Identities

Reciprocal identities relate functions like cosecant (csc θ = 1/sin θ) and cotangent (cot θ = cos θ/sin θ) to sine and cosine. Quotient identities express tangent and cotangent as ratios of sine and cosine, facilitating the manipulation and verification of trigonometric equations.
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Related Practice
Textbook Question

Simplify each expression. See Example 4.

cos² 2x - sin² 2x

Textbook Question

Verify that each equation is an identity.

(sin 2x)/(2sin x) = cos² (x/2) - sin² (x/2)

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Textbook Question

Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function of θ alone.

cos(90° + θ)

Textbook Question

Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function of θ alone.

cos(270° + θ)

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Textbook Question

Verify that each equation is an identity.

2 cos³ x - cos x = (cos² x - sin² x)/sec x

Textbook Question

Express each function as a trigonometric function of x. See Example 5.


cos 3x

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