Problem 52 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{1+\csc \theta}{\sec \theta}-\cot \theta=\cos \theta$$

Step-by-Step Solution

Answer

The given identity is indeed valid and simplifies to \(\cos \theta\). The algebraic solution is confirmed by a graphing utility

Step 1: Substitute Reciprocal Identities

Replace \(\csc \theta\) and \(\sec \theta\) in the given identity with their reciprocal identities: \(\frac{1 + \frac{1}{\sin \theta}}{\frac{1}{\cos \theta}} - \frac{\cos \theta}{\sin \theta}\)

Step 2: find Common Denominators and Simplify

Find a common denominator for the terms in the numerator and then simplify. The terms become \( \frac{\sin \theta + 1}{\sin \theta} \times \cos \theta - \frac{\cos \theta}{\sin \theta}\)

Step 3: Continue Simplifying

The expression simplifies to \((\cos \theta) \times (\frac{\sin \theta + 1}{\sin \theta} - 1)\)

Step 4: Distribute \( \cos \theta \)

Distribute the \(\cos \theta\) to get \(\cos \theta + \cos \theta - \cos \theta\)

Step 5: Final Simplification

Simplifying the last expressions gives \(\cos \theta\)

Key Concepts

Reciprocal Identities

Reciprocal identities are fundamental in trigonometry. They allow you to express the basic trigonometric functions in different ways, helping to simplify complex problems. For example, the cosecant (\(\csc\theta\) is the reciprocal of sine, expressed as \(\csc \theta = \frac{1}{\sin \theta}\). Similarly, the secant (\(\sec\theta\) is the reciprocal of cosine, or \(\sec \theta = \frac{1}{\cos \theta}\). These identities can be very useful when transforming trigonometric expressions. In the step-by-step solution, the first step was to replace \(\csc \theta\) and \(\sec \theta\) with their reciprocal identities. This replaced complicated terms with simpler fractions that could work together more easily. By understanding and using these reciprocal relationships, you can turn complex trigonometric expressions into manageable parts, setting the foundation for simplifying the expression in the following steps.

Simplification of Expressions

Simplifying trigonometric expressions is all about making them smaller or easier to understand without changing their value. Once reciprocal identities have been substituted, the next task is often to find common denominators. In this context, the common denominator ensures all fractions involved can be combined into a single coherent expression. In the step-by-step solution, we combined terms by finding common denominators and then proceeded to simplify where possible. By restructuring the expression to \(\frac{\sin \theta + 1}{\sin \theta} \times \cos \theta - \frac{\cos \theta}{\sin \theta}\) the problem became more straightforward to handle. Further simplification involved combining like terms and performing algebraic operations like distributing terms, which in this case was the \(\cos \theta\) over other terms. As seen in the exercise, simplification was completed when the expression resolved to \(\cos \theta\), matching the right side of the identity to prove it accurately.

Graphing Utilities

Graphing utilities are extremely helpful tools for visually verifying algebraic identities. They provide a graphical representation that can show the equality of two expressions over a range of values, which acts as a secondary check to algebraic solutions.Once you have simplified a trigonometric expression and verified its identity algebraically, using a graphing utility like a graphing calculator or software can help confirm your work. By graphing both sides of the identity equation, you can observe whether the graphs coincide over a specified interval of \(\theta\). If they do, it corroborates that the algebraic manipulations done were correct.For this exercise, after simplifying to \(\cos\theta\), graphing both \(\frac{1+\csc\theta}{\sec\theta}-\cot\theta\) and \(\cos \theta\) will help ensure that there is no graphical discrepancy between them across a range of values. This verification step provides additional confidence in the mathematical process and outcome.