Problem 147 - Calculus Volume 1

Question

For the following exercises, verify that each equation is an identity. $$\frac{\tan \theta \cot \theta}{\csc \theta}=\sin \theta$$

Step-by-Step Solution

Answer

The equation is an identity; both sides equal \( \sin \theta \).

Step 1: Understand Trigonometric Identities

The problem requires using trigonometric identities. Recall the basic identities: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), and \( \csc \theta = \frac{1}{\sin \theta} \). These will help transform and simplify the expression.

Step 2: Substitute Identities in the Left Side

Replace \( \tan \theta \) with \( \frac{\sin \theta}{\cos \theta} \) and \( \cot \theta \) with \( \frac{\cos \theta}{\sin \theta} \). The expression becomes: \[ \frac{\frac{\sin \theta}{\cos \theta} \times \frac{\cos \theta}{\sin \theta}}{\csc \theta} \]

Step 3: Simplify the Expression

Notice that \( \frac{\sin \theta}{\cos \theta} \times \frac{\cos \theta}{\sin \theta} = 1 \) because the cosine and sine terms cancel each other out. So the expression becomes \( \frac{1}{\csc \theta} \).

Step 4: Use Reciprocal Identity

Since \( \csc \theta = \frac{1}{\sin \theta} \), \( \frac{1}{\csc \theta} = \sin \theta \). So the left side of the equation becomes \( \sin \theta \).

Step 5: Verify the Equation

Compare the simplified left side, \( \sin \theta \), with the right side of the equation, which is also \( \sin \theta \). Since both sides are equal, the equation \( \frac{\tan \theta \cot \theta}{\csc \theta} = \sin \theta \) is verified to be an identity.

Key Concepts

Verifying Identities

Verifying trigonometric identities is like solving a puzzle. The aim is to prove that both sides of an equation are equal by transforming one side to match the other. In this exercise, we start with the identity \( \frac{\tan \theta \cot \theta}{\csc \theta} = \sin \theta \). Understanding this is crucial, as every trigonometric identity provides us a way to express trigonometric functions in terms of others. It's similar to using synonyms in language; they mean the same thing, even when they look different. Verifying identities involves using known identities to rewrite one side of the equation to see if it matches the other. This process helps strengthen your understanding of trigonometric relationships.

Trigonometric Simplification

Trigonometric simplification involves reducing complex trigonometric expressions to their simplest form. This often involves substituting one identity into another and canceling terms when possible. For instance, In our given problem, we start with \( \frac{\tan \theta \cot \theta}{\csc \theta} \). By substituting the known identities, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), we reduced the complex expression to \( \frac{1}{\csc \theta} \) after realizing the cancelation of the cosine and sine.

Reciprocal Identities

Reciprocal identities are trigonometric identities that involve the reciprocals of the basic trigonometric functions. These are helpful to know because they allow you to transform expressions, making it easier to simplify or verify identities. In our case, we used the reciprocal identity \( \csc \theta = \frac{1}{\sin \theta} \), which makes \( \frac{1}{\csc \theta} = \sin \theta \). Understanding these identities is essential in situations where you need to convert between functions, or when simplifying expressions that contain reciprocal terms.