Problem 65 - Algebra and Trigonometry with Analytic Geometry

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 as both sides are \(\sec \theta - \cos \theta\).

Step 1: Expand the Left-Hand Side

The given expression on the left-hand side is \((\cot \theta + \csc \theta)(\tan \theta - \sin \theta)\). To expand this expression, use the distributive property: \[\cot \theta \cdot \tan \theta - \cot \theta \cdot \sin \theta + \csc \theta \cdot \tan \theta - \csc \theta \cdot \sin \theta.\]

Step 2: Simplify Each Term

Start simplifying each part of the expanded expression:1. \(\cot \theta \cdot \tan \theta = (\frac{\cos \theta}{\sin \theta}) \cdot (\frac{\sin \theta}{\cos \theta}) = 1\).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. \(\csc \theta \cdot \sin \theta = \frac{1}{\sin \theta} \cdot \sin \theta = 1\).

Step 3: Combine the Simplified Terms

The expression now becomes:\[1 - \cos \theta + \sec \theta - 1.\]Combine like terms to further simplify the expression:\[-\cos \theta + \sec \theta.\]

Step 4: Verify Agreement with Right-Hand Side

The simplified left-hand side expression \(-\cos \theta + \sec \theta\) is equivalent to the right-hand side given, \(\sec \theta - \cos \theta\). As the transformed left-hand side matches the right-hand side, the identity is verified.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions of an angle, commonly used in the study of triangles and oscillations. Especially in this exercise, we deal with the following primary trigonometric functions:

• \( \sin \theta \): sine function, representing the ratio of the opposite side to the hypotenuse in a right triangle.
• \( \cos \theta \): cosine function, representing the ratio of the adjacent side to the hypotenuse.
• \( \tan \theta \): tangent function, which is the ratio of the sine to the cosine, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

In addition to these, this problem involves a few reciprocal trigonometric functions:

• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \): cotangent, the reciprocal of the tangent function.
• \( \csc \theta = \frac{1}{\sin \theta} \): cosecant, the reciprocal of sine.
• \( \sec \theta = \frac{1}{\cos \theta} \): secant, the reciprocal of cosine.

These reciprocal functions are crucial in transforming and simplifying expressions.

Distributive Property

The distributive property is a useful mathematical rule that allows us to multiply a single term across terms inside parentheses. It is particularly helpful when working with algebraic expressions. When dealing with the expression \((a+b)(c-d)\), the distributive property enables us to expand it to \( a\cdot c - a\cdot d + b\cdot c - b\cdot d \). In this problem, the distributive property lets us expand the expression \((\cot \theta + \csc \theta)(\tan \theta - \sin \theta)\) into individual terms. By distributing each part, we gain a clearer understanding, which is essential for further simplification. Applying this property in trigonometry helps simplify and verify identities.

Simplification of Expressions

Simplification involves reducing complex expressions into a simpler form, making mathematical problems easier to solve. In this exercise, each term from the expanded expression needs simplification.

• \( \cot \theta \cdot \tan \theta \) becomes \(1\) because it simplifies the division of equal factors.
• \( \cot \theta \cdot \sin \theta \) simplifies to \( \cos \theta \) as the \( \sin \theta \) terms cancel.
• \( \csc \theta \cdot \tan \theta \) converts to \( \sec \theta \) by dividing sine and multiplying with the reciprocal of cosine.
• \( \csc \theta \cdot \sin \theta \) simplifies down to \(1\) since they are reciprocals.

Simplifying expressions is crucial as it makes equations easier to work with and understand, leading to the final expression of the identity.

Verifying Identities

Verifying identities is about demonstrating that two different expressions are equal for all values within their domain. It's a way to prove the correctness of an equation or rule. In this exercise, the task is to show that the left-hand side, \((\cot \theta + \csc \theta)(\tan \theta - \sin \theta)\), can be simplified into the right-hand side, \(\sec \theta - \cos \theta\). After simplification, obtaining the same result as the right-hand side means the identity holds true. When verifying, this often involves:

• Simplifying complex expressions.
• Utilizing known trigonometric identities.
• Breaking down expressions using properties like the distributive property.

This ensures that expressions are accurate and mathematically valid across their range.