Problem 25 - Algebra 2 and Trigonometry

Question

In \(3-26,\) prove that each equation is an identity. $$ \frac{\csc \theta}{\sin \theta}-\cot ^{2} \theta=1 $$

Step-by-Step Solution

Answer

The equation is an identity: both sides simplify to 1.

Step 1: Start with the Left Side

Let's start with the left side of the equation: \( \frac{\csc \theta}{\sin \theta}-\cot ^{2} \theta \). First, we know that \( \csc \theta = \frac{1}{\sin \theta} \). Therefore, \( \frac{\csc \theta}{\sin \theta} = \frac{\frac{1}{\sin \theta}}{\sin \theta} = \frac{1}{\sin^2 \theta} \). So the left side becomes \( \frac{1}{\sin^2 \theta} - \cot^2 \theta \).

Step 2: Express Terms using Sine and Cosine

Continue by expressing \( \cot^2 \theta \) in terms of sine and cosine. Since \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), we have \( \cot^2 \theta = \left(\frac{\cos \theta}{\sin \theta}\right)^2 = \frac{\cos^2 \theta}{\sin^2 \theta} \). Thus, the expression becomes \( \frac{1}{\sin^2 \theta} - \frac{\cos^2 \theta}{\sin^2 \theta} \).

Step 3: Combine the Fractions

Since both terms have a common denominator \( \sin^2 \theta \), we can combine them into a single fraction: \( \frac{1 - \cos^2 \theta}{\sin^2 \theta} \).

Step 4: Use Pythagorean Identity

Apply the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), which gives \( 1 - \cos^2 \theta = \sin^2 \theta \). Substituting this into our fraction gives \( \frac{\sin^2 \theta}{\sin^2 \theta} \).

Step 5: Simplify the Expression

Simplify \( \frac{\sin^2 \theta}{\sin^2 \theta} \) to get 1, as any non-zero number divided by itself equals 1. This confirms that the left side of the equation simplifies to 1.

Step 6: Compare to the Right Side

The right side of the equation is 1. Since the left side also simplified to 1, we have shown that \( \frac{\csc \theta}{\sin \theta} - \cot^2 \theta = 1 \), confirming that the equation is indeed an identity.

Key Concepts

Understanding Cosecant

Cosecant is one of the six fundamental trigonometric functions. It is often abbreviated as "csc" and is the reciprocal of the sine function. In mathematical terms,

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

This means that whenever you have the sine of an angle, to find the cosecant, simply take the reciprocal of the sine value. Understanding this relationship is crucial when simplifying expressions or proving identities in trigonometry.
For example, let's say the sine of an angle \( \theta \) is 0.5. The cosecant of \( \theta \) would then be \( \csc \theta = \frac{1}{0.5} = 2 \).
This reciprocal relationship helps us transform and simplify trigonometric expressions, making it easier to handle in proofs and calculations.

Defining Cotangent

Cotangent, denoted as "cot," is another important trigonometric function. It is defined in terms of the sine and cosine functions. Specifically, it is the reciprocal of the tangent function.

• \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \)

Understanding cotangent involves recognizing that it expresses the ratio of cosine over sine.

In practical terms, let's consider an angle \( \theta \) where \( \cos \theta = \frac{3}{5} \) and \( \sin \theta = \frac{4}{5} \). Here, \( \cot \theta = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \).
Cotangent can be particularly useful in solving problems involving right triangles and circular functions, as it provides another way to express the tangent-related ratio.

Exploring Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry that connects sine and cosine, the principal trigonometric functions. It states:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity is derived from the Pythagorean theorem in a right triangle.

It's a powerful tool because it allows us to express sine in terms of cosine (or vice versa), and it helps in simplifying expressions involving squares of these functions. For instance, if you know \( \sin^2 \theta = 0.6 \), then \( \cos^2 \theta = 1 - 0.6 = 0.4 \).
The Pythagorean identity is especially useful in transforming complex trigonometric expressions into simpler forms, aiding in the proof of identities like the one in the original problem.

Sine and Cosine Relationships

Sine and cosine are deeply intertwined in trigonometry, and their relationship forms the basis for many trigonometric identities and transformations. They are often referred to together because of their complementary values. If you have an angle \( \theta \), the sine and cosine are complementary in the sense that:

• \( \sin(\frac{\pi}{2} - \theta) = \cos \theta \)
• \( \cos(\frac{\pi}{2} - \theta) = \sin \theta \)

This is known as the co-function identity, highlighting the symmetrical properties of these functions.

Moreover, sine and cosine can be used to convert between different forms of trigonometric functions, making it easier to prove identities and solve equations. For example, knowing that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) or \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) simplifies the solving process.
Recognizing these relationships is essential for a strong foundation in trigonometry.