Problem 54 - Precalculus: Functions and Graphs

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\cot \theta \sec \theta=\csc \theta$$

Step-by-Step Solution

Answer

The identity is verified: \( \cot \theta \sec \theta = \csc \theta \).

Step 1: Start with the Left-Hand Side

Begin with the given expression on the left-hand side: \( \cot \theta \sec \theta \). We will transform this expression into something that matches the right-hand side, \( \csc \theta \).

Step 2: Rewrite Trigonometric Functions

Recall the definitions: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \). Substitute these into the expression: \[ \cot \theta \sec \theta = \left( \frac{\cos \theta}{\sin \theta} \right) \left( \frac{1}{\cos \theta} \right). \]

Step 3: Simplify the Expression

Multiply the fractions: \[ \left( \frac{\cos \theta}{\sin \theta} \right) \left( \frac{1}{\cos \theta} \right) = \frac{\cos \theta}{\sin \theta} \cdot \frac{1}{\cos \theta} = \frac{1}{\sin \theta}. \]

Step 4: Recognize the Result

Recall the identity \( \csc \theta = \frac{1}{\sin \theta} \). Therefore, the expression \( \frac{1}{\sin \theta} \) simplifies to \( \csc \theta \), matching the right-hand side of the identity.

Key Concepts

Cotangent

The cotangent, often abbreviated as \( \cot \theta \), is one of the basic trigonometric functions. It's the reciprocal of the tangent function.

• Formula: \( \cot \theta = \frac{1}{\tan \theta} \)
• Relates as: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

This means cotangent represents the ratio of the adjacent side to the opposite side in a right triangle.
In terms of the unit circle, it's the ratio of the x-coordinate to the y-coordinate of a point.
This function is often used to simplify expressions and verify trigonometric identities.

Secant

The secant function, symbolized as \( \sec \theta \), is another fundamental trigonometric function. It's the reciprocal of the cosine function.

• Formula: \( \sec \theta = \frac{1}{\cos \theta} \)
• In terms of right triangles, it's the ratio of the hypotenuse to the adjacent side.

In a unit circle perspective, secant can be understood as the length of the line from the origin to the point where a tangent line touches the circle.
Secant is particularly useful for transforming an expression or a trigonometric identity to a simpler form.

Cosecant

The cosecant, or \( \csc \theta \), is another key trigonometric function. It's the reciprocal of the sine function.

• Formula: \( \csc \theta = \frac{1}{\sin \theta} \)
• In right triangle terms, it's the ratio of the hypotenuse to the opposite side.

In terms of the unit circle, it corresponds to the reciprocal of the y-coordinate of a point on the circle.
Cosecant is often used in verifying trigonometric identities like the one provided.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in the study of triangles and modeling periodic phenomena.The primary functions include:

• Sine \( (\sin \theta) \)
• Cosine \( (\cos \theta) \)
• Tangent \( (\tan \theta) \)

Their reciprocal functions include cotangent, secant, and cosecant.
These functions relate angles to ratios of sides in a right triangle, extending also to circular functions via the unit circle.Each function has specific identities and properties that allow complex expression simplification and transformation:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)
• \( \sec \theta = \frac{1}{\cos \theta} \)

These identities serve as the foundation for solving and simplifying trigonometric expressions.

Simplifying Expressions

Simplifying expressions is a critical skill in mathematics, useful in making complex calculations more manageable.Steps for simplifying trigonometric expressions include:

• Identify and substitute reciprocal functions.
• Utilize fundamental identities like \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Reduce fractions to their simplest form.

By following these steps, one can transform an expression, like \( \cot \theta \sec \theta \), into a simpler form like \( \csc \theta \).
Mastery of this process aids in verifying identities, solving equations, and understanding the deeper connections between trigonometric functions.