Skip to main content
Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblem 3.3.62
Chapter 3, Problem 3.3.62

In Exercises 59–68, verify each identity.
cos²(θ/2) = (sec θ + 1)/(2 sec θ)

Verified step by step guidance
1
Start by writing down the given identity clearly: \(\frac{\theta \sec^2 \theta}{2} + 1 = \frac{\cos^2 \theta}{2 \sec \theta}\).
Recall the definition of secant: \(\sec \theta = \frac{1}{\cos \theta}\). Use this to rewrite all secant terms in the identity in terms of cosine.
Rewrite the left-hand side (LHS) by substituting \(\sec^2 \theta\) with \(\frac{1}{\cos^2 \theta}\), so the LHS becomes \(\frac{\theta}{2 \cos^2 \theta} + 1\).
Rewrite the right-hand side (RHS) by substituting \(\sec \theta\) with \(\frac{1}{\cos \theta}\), so the RHS becomes \(\frac{\cos^2 \theta}{2 \times \frac{1}{\cos \theta}} = \frac{\cos^2 \theta \times \cos \theta}{2} = \frac{\cos^3 \theta}{2}\).
Simplify both sides as much as possible and then check if they are equal by manipulating the expressions algebraically, such as finding a common denominator or factoring, to verify the identity.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

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 of the equation simplify to the same expression using known identities and algebraic manipulation.
Recommended video:
5:32
Fundamental Trigonometric Identities

Secant and Cosine Relationship

Secant (sec θ) is the reciprocal of cosine (cos θ), defined as sec θ = 1/cos θ. Understanding this relationship is crucial for rewriting expressions and simplifying terms involving secant and cosine functions.
Recommended video:
6:22
Graphs of Secant and Cosecant Functions

Algebraic Manipulation in Trigonometry

Simplifying trigonometric expressions often requires algebraic skills such as factoring, finding common denominators, and combining fractions. These techniques help transform complex expressions into simpler or equivalent forms to verify identities.
Recommended video:
04:12
Algebraic Operations on Vectors
Related Practice
Textbook Question

In Exercises 53–62, solve each equation on the interval [0, 2𝝅). sin x + 2 sin x cos x = 0

5
views
Textbook Question

Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 2 cos² x + 3 cos x + 1 = 0

Textbook Question

In Exercises 53–62, solve each equation on the interval [0, 2𝝅). (2 cos x + √ 3) (2 sin x + 1) = 0

2
views
Textbook Question

In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression.

29. sin(5𝝅/12) cos(𝝅/4) - cos(5𝝅/12) sin(𝝅/4)

2
views
Textbook Question

In Exercises 53–62, solve each equation on the interval [0, 2𝝅). tan² x cos x = tan² x

7
views
Textbook Question

Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). tan(x/2) = √3

5
views