Ch. 5 - Trigonometric Identities

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problem 5.42

Chapter 6, Problem 5.42

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
1/ tan² α + cot α tan α

Verified step by step guidance

Recognize that \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \).

Rewrite \( \frac{1}{\tan^2 \alpha} \) as \( \cot^2 \alpha \) using the identity \( \cot \alpha = \frac{1}{\tan \alpha} \).

Substitute \( \cot^2 \alpha \) for \( \frac{1}{\tan^2 \alpha} \) in the expression.

Simplify \( \cot \alpha \tan \alpha \) to 1, since \( \cot \alpha = \frac{1}{\tan \alpha} \).

Combine the simplified terms: \( \cot^2 \alpha + 1 \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

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 that involve trigonometric functions and are true for all values of the variables involved. Fundamental identities, such as the Pythagorean identities, reciprocal identities, and quotient identities, are essential for simplifying trigonometric expressions. Understanding these identities allows students to manipulate and transform expressions effectively.

Reciprocal Functions

Reciprocal functions in trigonometry refer to the relationships between sine, cosine, tangent, and their reciprocals: cosecant, secant, and cotangent. For example, the tangent function is the reciprocal of cotangent, and this relationship is crucial when simplifying expressions. Recognizing these relationships helps in rewriting expressions in a more manageable form.

Simplification Techniques

Simplification techniques in trigonometry involve rewriting complex expressions into simpler forms using identities and algebraic manipulation. This may include factoring, combining like terms, or substituting equivalent expressions. Mastering these techniques is vital for solving trigonometric equations and understanding their behavior in various contexts.

Recommended video:

6:20

Example 6

Related Practice

Textbook Question

Verify that each equation is an identity.

(sec α + csc α) (cos α - sin α) = cot α - tan α

Textbook Question

The half-angle identity

tan A/2 = ± √[(1 - cosA)/(1 + cos A)]

can be used to find tan 22.5° = √(3 - 2√2), and the half-angle identity

tan A/2 = sin A/(1 + cos A)

can be used to find tan 22.5° = √2 - 1. Show that these answers are the same, without using a calculator. (Hint: If a > 0 and b > 0 and a² = b², then a = b.)

Textbook Question

Find sin θ.

cos θ = 5/6, θ in quadrant I

views

Textbook Question

Find the remaining five trigonometric functions of θ.

cos θ = 1/5, θ in quadrant I

Textbook Question

Find sinθ.

csc θ = -8/5

Textbook Question

Verify that each equation is an identity.