Problem 62 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 61-64: Either show that the equation is an identity or show that the equation is not an identity. $$ \frac{\tan ^{2} x}{\sec x-1}=\sec x $$

Step-by-Step Solution

Answer

The equation is not an identity.

Step 1: Understand the Trigonometric Identities Involved

We need to determine if \( \frac{\tan ^{2} x}{\sec x-1}=\sec x \) is an identity. Recall the trigonometric identities: \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \). This means \( \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \) and \( \sec x - 1 = \frac{1 - \cos x}{\cos x} \). We'll use these identities to manipulate both sides of the equation.

Step 2: Simplify the Left Side of the Equation

Substitute \( \tan^2 x \) and \( \sec x - 1 \) into the left side of the equation:\[\frac{\tan^2 x}{\sec x - 1} = \frac{\frac{\sin^2 x}{\cos^2 x}}{\frac{1 - \cos x}{\cos x}} = \frac{\sin^2 x}{\cos^2 x} \times \frac{\cos x}{1 - \cos x} = \frac{\sin^2 x \cdot \cos x}{\cos^2 x \cdot (1 - \cos x)}.\] Reduce the \( \cos^2 x \) in the denominator by \( \cos x \) to get:\[\frac{\sin^2 x}{\cos x (1 - \cos x)}.\]

Step 3: Simplify the Right Side of the Equation

The right side of our original equation is simply \( \sec x = \frac{1}{\cos x} \).

Step 4: Compare Both Sides

To determine if the equation is an identity, both sides of the equation must be equivalent for all values of \( x \). Compare the left side \( \frac{\sin^2 x}{\cos x (1 - \cos x)} \) and right side \( \frac{1}{\cos x} \).Rewrite \( \sin^2 x = 1 - \cos^2 x \), giving:\[\frac{1 - \cos^2 x}{\cos x (1 - \cos x)}.\] Separate the expression to:\[\frac{(1 - \cos x)(1 + \cos x)}{\cos x (1 - \cos x)}.\] Cancel \( (1 - \cos x) \) in the numerator and denominator:\[\frac{1 + \cos x}{\cos x} eq \frac{1}{\cos x}.\]

Step 5: Conclusion

Since \( \frac{1 + \cos x}{\cos x} eq \frac{1}{\cos x} \), the equation \( \frac{\tan^2 x}{\sec x - 1} = \sec x \) is not an identity. This demonstrates that the two sides of the equation are not equal for all \( x \).

Key Concepts

Understanding Trigonometric Equations

Trigonometric equations are mathematical statements that involve trigonometric functions like sine, cosine, tangent, etc. These equations usually seek to find the values of angles (or variable) that make the statement true. A trigonometric identity, on the other hand, holds true for all values within the domain of the variable.

Solving trigonometric equations often involves familiarizing yourself with basic identities such as:

• \(\sin^2 x + \cos^2 x = 1\)
• \(\tan x = \frac{\sin x}{\cos x}\)
• \(\sec x = \frac{1}{\cos x}\)

These identities help us transform one side of a trigonometric equation to match the other or to simplify complex expressions. Understanding how to manipulate these identities is key to solving or proving trigonometric equations.

Harnessing Trigonometric Simplification

Trigonometric simplification is the process of making a trigonometric expression simpler or easier to work with. This often involves applying identities and algebraic manipulations to reduce complexity.

Take the expression \(\frac{\tan^2 x}{\sec x-1}\), as an example. To simplify:

• Use the identity \(\tan^2 x = \frac{\sin^2 x}{\cos^2 x}\).
• Recognize that \(\sec x - 1\) simplifies to \(\frac{1-\cos x}{\cos x}\).
• Combine these to get: \(\frac{\sin^2 x}{\cos^2 x} \times \frac{\cos x}{1 - \cos x}\).

By reducing the expression, you can better compare it to the simpler right side of our equation, which in this case is \(\sec x = \frac{1}{\cos x}\). This process is essential for validating whether a given trigonometric equation is an identity, meaning it would hold for all variables.

Exploring Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent play a crucial role in numerous areas of mathematics and are often the core elements of trigonometric equations. Each of these functions is derived from the relationships within a right triangle or their respective points on the unit circle.

These functions are interconnected through identities:

• \(\tan x = \frac{\sin x}{\cos x}\) connects tangent, sine, and cosine.
• Reciprocal identities such as \(\sec x = \frac{1}{\cos x}\) and \(\csc x = \frac{1}{\sin x}\) display relationships between angles and their cosine or sine values, respectively.

Knowledge of these functions and identities is imperative in simplifying and solving trigonometric equations and, as demonstrated, is pivotal in proofs and transformations needed for complex mathematical problems. By mastering these components, you set a strong foundation for advanced learning in trigonometry and beyond.