Problem 65 - Algebra 2

Question

\(\begin{array}{ll}{\text { How can you express } \csc ^{2} \theta-2 \cot ^{2} \theta \text { in terms of } \sin \theta \text { and } \cos \theta ?} \\\ {\mathrm{F} \cdot \frac{1-2 \cos ^{2} \theta}{\sin ^{2} \theta}} & {\text { G. } \frac{1-2 \sin ^{2} \theta}{\sin ^{2} \theta}} \\ {\mathrm{H} \cdot \sin ^{2} \theta-2 \cos ^{2} \theta} & {\text { J. } \frac{1}{\sin ^{2} \theta}-\frac{2}{\tan ^{2} \theta}}\end{array}\)

Step-by-Step Solution

Answer

F: \(\frac{1-2\cos^2\theta}{\sin^2\theta}\)

Step 1: Convert to Sine and Cosine

Rewrite \(\csc^2\theta\) as \((1 / \sin^2\theta)\) and \(\cot^2\theta\) as \((\cos^2\theta / \sin^2\theta)\). The equation becomes \((1 / \sin^2\theta) - 2(\cos^2\theta / \sin^2\theta)\).

Step 2: Simplify the Equation

The equation from Step 1 has a common denominator, \(\sin^2\theta\). By combining the terms, the equation simplifies to \(1 / \sin^2\theta - 2\cos^2\theta / \sin^2\theta\). This simplifies to \((1 - 2\cos^2\theta) / \sin^2\theta\) (Place the numerator under the common denominator \(\sin^2\theta\)).

Step 3: Find the Correct Answer

The results matches with answer choice F, \((1 - 2\cos^2\theta / \sin^2\theta)\), therefore this is the correct answer.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is one of the fundamental trigonometric functions often used in various mathematical calculations. It is defined as the reciprocal of the sine function, which means that \( \csc \theta = \frac{1}{\sin \theta} \). This relationship makes the cosecant function particularly useful when dealing with angles and lengths in right-angled triangles.

In the context of trigonometric identities, the cosecant function plays a crucial role in expressions that involve division by the sine of an angle. By understanding that \( \csc \theta \) is simply \( \frac{1}{\sin \theta} \), solving expressions like \( \csc^2 \theta \) becomes straightforward, as it can be rewritten as \( \left(\frac{1}{\sin^2 \theta}\right) \). This simplification is essential for breaking down complex trigonometric identities into simpler forms.

Cotangent Function

The cotangent function, represented as \( \cot \theta \), is another trigonometric function that holds a significant place in trigonometrical computations. Cotangent is defined as the reciprocal of the tangent function: \( \cot \theta = \frac{1}{\tan \theta} \). Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), it follows that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Due to its definition, the cotangent function is particularly useful in identities that simplify expressions involving divisions by the tangent function. It also finds importance in solving various trigonometric equations and identities, often being a part of simplification processes. For instance, \( \cot^2 \theta \) which can be expressed as \( \left(\frac{\cos^2 \theta}{\sin^2 \theta}\right) \), is crucial in expressing complicated trigonometric equations in terms of sine and cosine, as seen in the original exercise.

Trigonometric Simplification

Trigonometric simplification is a powerful tool in mathematics. It allows for the reduction of complex trigonometric expressions to simpler and more manageable forms. The process often involves rewriting trigonometric functions in terms of sine and cosine, which are considered the most fundamental trigonometric functions.

When simplifying expressions such as \( \csc^2 \theta - 2 \cot^2 \theta \), one method is to convert all terms to sine and cosine functions. This is because many basic trigonometric identities are based on these two functions, like \( \csc \theta = \frac{1}{\sin \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). This conversion simplifies recognition of common denominators and facilitates easier combination of terms, ultimately leading to a cleaner and more understandable result.

Sine and Cosine Relationships

The sine and cosine functions form the backbone of trigonometric calculations, with many trigonometric identities revolving around these two functions. The sine function, \( \sin \theta \), and the cosine function, \( \cos \theta \), represent the ratios of sides in a right triangle and are intimately linked through the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \).

This relationship allows for transformations and substitutions in trigonometric expressions to simplify them. In the context of our exercise, replacing \( \csc \theta \) and \( \cot \theta \) with their sine and cosine equivalents enables us to apply this identity to achieve a simplified form. Engaging with these relationships often leads to finding common factors and simplifying complex expressions, allowing for practical applications of trigonometry in both theoretical and real-world problems.