Problem 89 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \(\frac{\cos \theta}{\sin \theta \cot \theta}=1\)

Step-by-Step Solution

Answer

Yes, the equation is an identity.

Step 1: Understand the Equation

We are given the equation \( \frac{\cos \theta}{\sin \theta \cot \theta}=1 \). Our task is to verify whether this equation is an identity, which means it must hold for all permissible values of \( \theta \).

Step 2: Simplify the Expression

Recall that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Substitute this into the equation: \( \frac{\cos \theta}{\sin \theta \cdot \frac{\cos \theta}{\sin \theta}} \).

Step 3: Resolve the Denominator

Simplify the expression in the denominator: \( \sin \theta \cdot \frac{\cos \theta}{\sin \theta} = \cos \theta \). Thus the equation becomes \( \frac{\cos \theta}{\cos \theta} \).

Step 4: Simplify the Fraction

The fraction \( \frac{\cos \theta}{\cos \theta} \) simplifies to 1 (provided \( \cos \theta eq 0 \)).

Step 5: Verify the Identity

We have shown that \( \frac{\cos \theta}{\sin \theta \cot \theta} = 1 \) simplifies correctly to 1. Therefore, the original equation is an identity for all values of \( \theta \) where \( \cos \theta eq 0 \).

Key Concepts

Simplification

Simplifying expressions in trigonometry is a crucial skill that helps us understand equations more clearly and reach the solution efficiently. In our exercise,the first step is to simplify the given expression \(\frac{\cos \theta}{\sin \theta \cot \theta}=1\). Understanding how each trigonometric function relates to others enables us to manipulate the equation easily.
To start, recognize the trigonometric identity for the cotangent: \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). By substituting this identity into the denominator of our exercise, the expression becomes easier to navigate through:

• Convert \(\sin \theta \cdot \cot \theta\) to \(\sin \theta \cdot \frac{\cos \theta}{\sin \theta}\).
• Simplify the multiplication, noticing that \(\sin \theta\) cancels out with \(\sin \theta\) in the denominator, leaving you with \(\cos \theta\).

By simplifying, you change the equation to \(\frac{\cos \theta}{\cos \theta}\), which directly reduces to 1, ensuring that the equation holds true.

Cotangent

The cotangent function is one of the six fundamental trigonometric functions, often abbreviated as \(\cot\). It constitutes a reciprocal relationship with another function.
Specifically, \(\cot \theta\) is defined as the reciprocal of the tangent function, which means \(\cot \theta = \frac{1}{\tan \theta}\). In terms of sine and cosine, the relationship becomes \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). This identity is particularly useful in solving trigonometric equations, especially those involving products or fractions.
When dealing with trigonometric expressions, converting cotangent using its identity often simplifies complex relationships. You use this identity to simplify expressions like the one in the exercise, breaking down terms into simpler components that are easier to work with.

Verifying Identities

Verifying trigonometric identities involves demonstrating that two sides of an equation are equivalent. This means showing that the equation is true for all values within its domain. In the given exercise, we started with the expression \(\frac{\cos \theta}{\sin \theta \cot \theta}=1\) and simplified it to verify its identity.
The verification process often involves:

• Using known identities, such as \(\cot \theta = \frac{\cos \theta}{\sin \theta}\), to transform parts of the equation.
• Simplifying the expression step by step, ensuring each step is backed by logical trigonometric relationships.
• Confirming that the final simplified form of the equation is equivalent to a universal truth, such as 1 or another simple trigonometric result.

In the end, we've confirmed that our original equation holds as a true identity across allowable values of \(\theta\), thus successfully verifying its correctness.