Problem 17 - Precalculus with Limits

Question

In Exercises 9-50, verify the identity \( \dfrac{\tan^2 \theta}{\sec \theta} = \sin \theta \tan \theta \)

Step-by-Step Solution

Answer

The identity is verified as \( \frac{\tan^2 \theta}{\sec \theta} = \sin \theta \tan \theta \).

Step 1: Extract the definition

Write down the definitions of \( \tan \) and \( \sec \) in terms of sine and cosine. So, we have \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \).

Step 2: Substitute these definitions in the expression

Substitute these definitions in the given expression. So, you have \( \frac{\tan^2 \theta}{\sec \theta} = \frac {\left( \frac{\sin \theta}{\cos \theta} \right)^2 }{\frac{1}{\cos \theta}} \).

Step 3: Simplify the equation

On simplifying above equation, we get \( \frac{\sin^2 \theta}{\cos^2 \theta} * \cos \theta = \frac{\sin^2 \theta }{\cos \theta} \).

Step 4: Further simplification

This will result in \( \sin \theta * \frac{\sin \theta}{\cos \theta} = \sin \theta * \tan \theta \). Hence, proved.

Step 5: Final Step

Check, does our answer match with given right hand side of equation, to show the identity is true. So, \( \sin \theta \tan \theta = \sin \theta \tan \theta \).

Key Concepts

Sine and Cosine Functions

Understanding the sine and cosine functions is crucial for solving many trigonometric problems, including the verification of trigonometric identities. The sine function, denoted as \( \sin \theta \), represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. Similarly, the cosine function, represented as \( \cos \theta \), is the ratio of the length of the adjacent side to the hypotenuse. These functions are fundamental in expressing other trigonometric functions.
For instance:

• The tangent function \( \tan \theta \) can be expressed as \( \frac{\sin \theta}{\cos \theta} \)
• The secant function \( \sec \theta \) is \( \frac{1}{\cos \theta} \)

By converting expressions into terms of sine and cosine, you simplify the process of verifying identities. This conversion helps us see common patterns and ultimately resolve complicated expressions more easily, as seen by transforming \( \tan \) and \( \sec \) in the given identity.

Trigonometric Equations

Trigonometric equations form the backbone of solving problems related to angles and their identities. These equations can often seem daunting but simplify significantly when expressed in fundamental terms using sine and cosine. Consider the equation given in the exercise. The goal is to verify: \[ \frac{\tan^2 \theta}{\sec \theta} = \sin \theta \tan \theta \] Substituting \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \) into the equation, transforms it into terms that can be simplified using basic algebraic operations.

• First, square the tangent term: \( \tan^2 \theta = \left( \frac{\sin \theta}{\cos \theta} \right)^2 \)
• Next, divide by the secant term: \( \frac{1}{\cos \theta} \)

The simplifications lead to easier management of the components and lay the groundwork for verifying trigonometric identities, reinforcing that expressing functions in terms of sine and cosine often simplifies these equations.

Angle Identities

Angle identities help us simplify and verify equations involving angles through known patterns and relationships. In this exercise, recognizing the identity of the tangent and secant functions in terms of sine and cosine is key.

• The given identity simplifies to \( \frac{\sin^2 \theta}{\cos \theta} \)
• Which then resolves to \( \sin \theta \cdot \tan \theta \)

This transformation stems from recognizing that the tangent squared over the secant can be represented using sine and cosine functions, following the identity \( \sin \theta \cdot \tan \theta = \sin \theta \cdot \frac{\sin \theta}{\cos \theta} \). Recognizing such identities allows you to simplify and verify complex expressions effectively.
When tackling trigonometric proofs, familiarity with these angle identities helps in observing equivalences and ensuring that transformations correspond with known identities. This skill is essential in simplifying trigonometric expressions and proving more involved identities, such as the one being verified in the exercise.