Problem 65 - Precalculus: Functions and Graphs

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$(\cot \theta+\csc \theta)(\tan \theta-\sin \theta)=\sec \theta-\cos \theta$$

Step-by-Step Solution

Answer

The identity is verified: LHS simplifies to the RHS.

Step 1: Expand Left-Hand Side

We need to expand the left-hand side expression \((\cot \theta + \csc \theta)(\tan \theta - \sin \theta)\). Distribute the terms:\[ \cot \theta \cdot \tan \theta - \cot \theta \cdot \sin \theta + \csc \theta \cdot \tan \theta - \csc \theta \cdot \sin \theta \]

Step 2: Simplify Each Term

Simplify each of the terms obtained:1. \( \cot \theta \cdot \tan \theta = 1 \) because \( \cot \theta = \frac{1}{\tan \theta} \).2. \( \cot \theta \cdot \sin \theta = \frac{\cos \theta}{\sin \theta} \cdot \sin \theta = \cos \theta \).3. \( \csc \theta \cdot \tan \theta = \frac{1}{\sin \theta} \cdot \frac{\sin \theta}{\cos \theta} = \frac{1}{\cos \theta} = \sec \theta \).4. \( \csc \theta \cdot \sin \theta = 1 \) because \( \csc \theta = \frac{1}{\sin \theta} \).Substitute the simplified terms back into the expanded expression:\[ 1 - \cos \theta + \sec \theta - 1 \]

Step 3: Combine Like Terms

Combine the like terms from the expression:\[ 1 - 1 - \cos \theta + \sec \theta = \sec \theta - \cos \theta \]

Step 4: Conclusion

The transformed expression \( \sec \theta - \cos \theta \) matches the right-hand side of the identity given in the exercise, verifying the identity is correct.

Key Concepts

Cotangent

Understanding the cotangent function is essential when working with trigonometric identities. The cotangent, denoted as \( \cot \theta \), is defined as the reciprocal of the tangent function.
Mathematically, \( \cot \theta = \frac{1}{\tan \theta} \), or equivalently, \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
This relationship shows how the cotangent is directly related to both sine and cosine.Key Points about Cotangent:

• Cotangent is the ratio of the adjacent side to the opposite side in a right triangle.
• It is undefined when \( \sin \theta = 0 \), due to division by zero.
• In the context of unit circles, cotangent values repeat every \( \pi \), making their periodicity \( \pi \).

To verify trigonometric identities, knowing how to express cotangent in terms of sine and cosine is very useful. As seen in the exercise solution, substituting \( \cot \theta \) with \( \frac{1}{\tan \theta} \) simplifies multiplication with \( \tan \theta \) to 1, which is a frequently used simplification step.

Cosecant

The cosecant function, denoted as \( \csc \theta \), is crucial for understanding complementary trigonometric identities. Cosecant is the reciprocal of the sine function. This means \( \csc \theta = \frac{1}{\sin \theta} \).
By using this definition, we can transform trigonometric expressions in useful ways, particularly in combination with sine.Key Points about Cosecant:

• It represents the hypotenuse divided by the opposite side in a right triangle context.
• It is undefined where \( \sin \theta = 0 \), causing division by zero issues at multiples of \( \pi \).
• The period of the cosecant function is \( 2\pi \), aligning with the sine function.

In the context of the example problem, expressing \( \csc \theta \) in relation to sine aids in simplification. Particularly, when pairing \( \csc \theta \cdot \sin \theta \), the result simplifies directly to 1, showcasing its utility in verification processes.

Secant

The secant function, symbolized as \( \sec \theta \), is often utilized in transforming expressions involving cosine. Secant is defined as the reciprocal of the cosine, which gives us \( \sec \theta = \frac{1}{\cos \theta} \).
Understanding this reciprocal relationship helps in simplifying trigonometric identities effectively.Key Points about Secant:

• Secant relates to the hypotenuse over the adjacent side in a right triangle.
• This function is undefined when \( \cos \theta = 0 \).
• It shares a period of \( 2\pi \) with the cosine function.

In the exercise solution provided, identifying \( \sec \theta \) allows for transformations that help verify the identity. Offering simplification when coupled with \( \cos \theta \), the expression \( \sec \theta - \cos \theta \) forms the right side of the original identity after simplification.

Tangent and Sine

Tangent (\( \tan \theta \)) and sine (\( \sin \theta \)) are two cornerstone trigonometric functions that often work hand-in-hand in identities.Tangent Function:

• Defined as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
• Represents the ratio of the opposite side to the adjacent side in right triangles.
• Tangent is periodic, with a period of \( \pi \).

When simplifying identities, recognizing \( \tan \theta \) as \( \frac{\sin \theta}{\cos \theta} \) can significantly ease the process.Sine Function:

• Expressed as \( \sin \theta \), it describes the y-coordinate on the unit circle.
• Linked directly to the opposite side in a triangle compared to the hypotenuse.
• Sine has a period of \( 2\pi \), which defines how often it repeats.

In the provided solution, using the definitions and relationships for tangent directly connects to simplifying and transforming the expression. The interplay between tangent and sine allows for multiplication and division operations that drive simplification towards verifying complex identities.