Problem 21 - Algebra 2 and Trigonometry

Question

In \(15-22,\) write each given expression in terms of sine and cosine and express the result in simplest form. \(\csc ^{2} \theta-\frac{\cot \theta}{\tan \theta}\)

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Express Cosecant and Cotangent

Recall the definitions: \(\csc \theta = \frac{1}{\sin \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). Therefore, \(\csc^2 \theta = \left(\frac{1}{\sin \theta}\right)^2 = \frac{1}{\sin^2 \theta}\).

Step 2: Express Tangent

Recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Therefore, \(\frac{\cot \theta}{\tan \theta} = \frac{\frac{\cos \theta}{\sin \theta}}{\frac{\sin \theta}{\cos \theta}}\).

Step 3: Simplify the Fraction

Simplify \(\frac{\frac{\cos \theta}{\sin \theta}}{\frac{\sin \theta}{\cos \theta}}\). Multiply numerators and denominators: \(\frac{\cos \theta}{\sin \theta} \times \frac{\cos \theta}{\sin \theta} = \frac{\cos^2 \theta}{\sin^2 \theta}\).

Step 4: Substitute into Original Expression

Substitute back into the original expression: \(\csc^2 \theta - \frac{\cot \theta}{\tan \theta}\), which becomes: \(\frac{1}{\sin^2 \theta} - \frac{\cos^2 \theta}{\sin^2 \theta}\).

Step 5: Combine and Simplify

Combine the fractions: \(\frac{1}{\sin^2 \theta} - \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{1 - \cos^2 \theta}{\sin^2 \theta}\). Use the Pythagorean identity: \(1 - \cos^2 \theta = \sin^2 \theta\). Thus, the expression simplifies to \(\frac{\sin^2 \theta}{\sin^2 \theta} = 1\).

Key Concepts

Cosecant and Cotangent

Cosecant and Cotangent are two of the lesser-known trigonometric functions, but understanding them is crucial to grasping more complex trigonometric identities. First, let's talk about cosecant, denoted as \( \csc \theta \). It is the reciprocal of the sine function. This means that \( \csc \theta = \frac{1}{\sin \theta} \). It comes in handy, especially when sine values approach zero, as it will give a clearer view of the angle's behavior under such conditions.

Similarly, cotangent, written as \( \cot \theta \), is the reciprocal of the tangent function. This translates to \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Knowing cotangent provides valuable insights into angles where cosine is equal to sine, or where the tangent is approaching infinite values.

Using these reciprocal relationships, you can transform expressions like \( \csc^2 \theta - \frac{\cot \theta}{\tan \theta} \) into expressions that are easier to simplify by rewriting them in terms of sine and cosine functions. This foundational understanding paves the way for tackling more intricate trigonometric identities and equations.

Tangent simplification

Simplifying trigonometric expressions often requires a keen understanding of the tangent function and its relationship with sine and cosine. Tangent, denoted as \( \tan \theta \), is simply the ratio of sine to cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This relationship becomes particularly useful when dealing with complex fractions that incorporate multiple trigonometric functions.

In our exercise, understanding this relationship allows us to handle the expression \( \frac{\cot \theta}{\tan \theta} \). By expressing cotangent and tangent in terms of sine and cosine, we have \( \frac{\frac{\cos \theta}{\sin \theta}}{\frac{\sin \theta}{\cos \theta}} \). Simplifying this fraction involves multiplying by the reciprocal, essentially flipping and multiplying the terms to cancel out the sines and cosines, resulting in \( \frac{\cos^2 \theta}{\sin^2 \theta} \).

This step is crucial in breaking down more complicated trigonometric expressions into simpler components, making further simplification or integration much smoother.

Pythagorean identity

The Pythagorean identity is one of the cornerstones of trigonometry. It asserts that for any angle \( \theta \), the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) holds true. This simply means that the squares of the sine and cosine of an angle add up to one.

In the context of our exercise, this powerful identity aids in further simplification. After transforming and simplifying the expression \( \csc^2 \theta - \frac{\cot \theta}{\tan \theta} \) to \( \frac{1}{\sin^2 \theta} - \frac{\cos^2 \theta}{\sin^2 \theta} \), you can combine these fractions into \( \frac{1 - \cos^2 \theta}{\sin^2 \theta} \).

When you apply the Pythagorean identity \( 1 - \cos^2 \theta = \sin^2 \theta \), the expression simplifies dramatically to \( \frac{\sin^2 \theta}{\sin^2 \theta} = 1 \). This reveals that all the complexity boils down to a simple and elegant outcome due to the Pythagorean identity, showcasing its power in simplifying trigonometric expressions.