Problem 147 - Calculus Volume 1

Question

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 because both sides simplify to \( \sin \theta \).

Step 1: Simplify the Left Side

We start with the left side of the equation: \[ \frac{\tan \theta \cot \theta}{\csc \theta}. \]First, use the trigonometric identities: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).So, \( \tan \theta \cot \theta = \frac{\sin \theta}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta} = 1 \). Thus, we have: \[ \frac{1}{\csc \theta}. \]

Step 2: Convert Cosecant

Now, convert \( \csc \theta \) using the identity \( \csc \theta = \frac{1}{\sin \theta} \). So, the expression becomes:\[ \frac{1}{\frac{1}{\sin \theta}} = \sin \theta. \]

Step 3: Verify Equality

The right side of the original equation is already \( \sin \theta \). Both sides of the equation simplify to \( \sin \theta \), confirming that the equation is an identity.

Key Concepts

Simplifying Trigonometric Expressions

Trigonometric expressions often appear complex at first glance. To simplify them, we frequently use trigonometric identities—essentially, relationships connecting the basic trigonometric functions such as sine, cosine, and tangent. Consider our problem where the expression starts as \( \frac{\tan \theta \cot \theta}{\csc \theta} \). To simplify this, we break down each function using identities:

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

By multiplying these, \( \tan \theta \cot \theta = \frac{\sin \theta}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta} \), which simplifies to 1, making the expression \( \frac{1}{\csc \theta} \). Breaking complex expressions with known identities is a powerful tool to simplify, turning them into more manageable forms.

Verification of Identities

Verifying trig identities involves proving that both sides of an equation are equal through logical steps. Here, we began with the expression \( \frac{\tan \theta \cot \theta}{\csc \theta} \) and simplified it to \( \sin \theta \). In the process:

• Simplify the expression using identities like \( \csc \theta = \frac{1}{\sin \theta} \).
• Result becomes \( \sin \theta \), which matches the equation's right side.

Both sides now read \( \sin \theta \), confirming the equation is an identity. This method demonstrates that despite the initial complexity, the expression reduces directly to the same form, making verification straightforward. Knowing common trigonometric identities and rules enables more abstract forms to unveil their equivalent, simpler identities.

Basic Trigonometric Functions

Trigonometric functions form the backbone of trigonometry and relate the angles of triangles to side ratios. Here is a brief rundown of some of the basic functions:

• Sine (sin): Ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine (cos): Ratio of the adjacent side to the hypotenuse.
• Tangent (tan): Ratio of the sine to the cosine, or opposite over adjacent sides, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
• Cotangent (cot): Reciprocal of tangent, or adjacent over opposite, \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
• Cosecant (csc): Reciprocal of sine, \( \csc \theta = \frac{1}{\sin \theta} \).

Understanding these basic trigonometric functions and their relationships is crucial because they allow us to manipulate and simplify expressions, enabling further exploration and application in various mathematical contexts. Knowledge of these functions serves as a toolkit for simplifying expressions or verifying identities.