Problem 53 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The identity \( \frac{\csc \theta}{\sec \theta} = \cot \theta \) is verified.

Step 1: Understand the Given Identity

We are given the identity \( \frac{\csc \theta}{\sec \theta} = \cot \theta \). Our task is to verify this by transforming the left-hand side (LHS) into the right-hand side (RHS).

Step 2: Recall Trigonometric Identities

Recall that \( \csc \theta = \frac{1}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \). We will use these identities to substitute into the LHS.

Step 3: Substitute Trigonometric Identities into LHS

Substitute \( \csc \theta = \frac{1}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \) into the LHS. We get:\[\frac{\csc \theta}{\sec \theta} = \frac{\frac{1}{\sin \theta}}{\frac{1}{\cos \theta}}.\]

Step 4: Simplify the Complex Fraction

To simplify \( \frac{\frac{1}{\sin \theta}}{\frac{1}{\cos \theta}} \), multiply the numerator by the reciprocal of the denominator:\[= \frac{1}{\sin \theta} \times \frac{\cos \theta}{1} = \frac{\cos \theta}{\sin \theta}.\]

Step 5: Recognize the Simplified Expression

The expression \( \frac{\cos \theta}{\sin \theta} \) can be recognized as \( \cot \theta \), since \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Thus, we have shown:\[\frac{\csc \theta}{\sec \theta} = \cot \theta.\] This completes the verification of the identity.

Key Concepts

Understanding Cosecant (csc)

Cosecant is a fundamental trigonometric function linked to the sine function. It is defined as the reciprocal of sine. The formula to represent cosecant is given by:

• \( \csc \theta = \frac{1}{\sin \theta} \)

This means that if you know the value of \( \sin \theta \), you can find \( \csc \theta \) by taking its reciprocal.
Cosecant is often useful in problems involving right-angled triangles or trigonometric identities because it helps express relationships opposite the hypotenuse.
In the given identity verification task, we use \( \csc \theta \) to transform the expression on the left-hand side.

Exploring Secant (sec)

Secant is another trigonometric function, directly related to the cosine function. It serves as the reciprocal of cosine. The mathematical representation is:

• \( \sec \theta = \frac{1}{\cos \theta} \)

This relationship allows you to find the secant by dividing 1 by the value of \( \cos \theta \).
Secant is particularly important in trigonometry for solving equations where cosine might not directly give the required result.
In our exercise, we replaced \( \sec \theta \) with its reciprocal form to simplify the fraction \( \frac{\csc \theta}{\sec \theta} \), aiding in the transformation process of the identity.

Decoding Cotangent (cot)

Cotangent is expressed as the reciprocal of the tangent function or as the ratio of cosine over sine. Its formula is known as:

• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

Understanding cotangent is essential in trigonometry since it helps in the simplification of trigonometric identities, much like in the present problem.
In verifying the identity \( \frac{\csc \theta}{\sec \theta} = \cot \theta \), the main goal was to arrive at \( \frac{\cos \theta}{\sin \theta} \) on the left-hand side by simplifying the given expression.
Once simplified, the left-hand side of the identity perfectly matches \( \cot \theta \), proving the identity true.