Problem 24 - Precalculus: Functions and Graphs

Question

Verify the identity by transforming the left hand side into the right-hand side. $$\frac{\sec (-x)}{\tan (-x)}=-\csc x$$

Step-by-Step Solution

Answer

The identity is verified: both sides equal \(-\csc x\).

Step 1: Use Even/Odd Identities

The identities for even and odd functions state that \( an(-x) = - an(x)\) and \( ext{sec}(-x) = \text{sec}(x)\). Applying these to the left side of the equation gives: \(\frac{\sec(-x)}{\tan(-x)} = \frac{\sec(x)}{-\tan(x)}\).

Step 2: Rewrite the Expression

Now rewrite the expression from Step 1: \(\frac{\sec(x)}{-\tan(x)} = -\frac{\sec(x)}{\tan(x)}\).

Step 3: Express in Terms of Sine and Cosine

Recall the definitions: \(\sec(x) = \frac{1}{\cos(x)}\) and \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). Substitute these into the expression: \(-\frac{\sec(x)}{\tan(x)} = -\frac{\frac{1}{\cos(x)}}{\frac{\sin(x)}{\cos(x)}}\).

Step 4: Simplify the Expression

To simplify \(-\frac{\frac{1}{\cos(x)}}{\frac{\sin(x)}{\cos(x)}}\), multiply numerator and denominator by \(\cos(x)\), resulting in \(-\frac{1}{\sin(x)}\).

Step 5: Recognize the Identity

Recall that \(\csc(x) = \frac{1}{\sin(x)}\). Therefore, \(-\frac{1}{\sin(x)} = -\csc(x)\), which matches the right-hand side of the original identity.

Key Concepts

Even and Odd Functions

In trigonometry, understanding the nature of even and odd functions helps in transforming complex expressions. An even function is one where symmetrical behavior around the y-axis occurs, such that \( f(-x) = f(x) \). For trigonometric functions, the secant function, \( \sec(x) \), is even: \( \sec(-x) = \sec(x) \). This cultural consistency can simplify identities in trigonometric equations.
Odd functions, on the other hand, exhibit point symmetry around the origin, defined by \( f(-x) = -f(x) \). The tangent function, \( \tan(x) \), is odd, thus \( \tan(-x) = -\tan(x) \). Recognizing these properties allows us to make initial transformations in trigonometric identities, drastically simplifying the left-hand side by changing it from \( \frac{\sec(-x)}{\tan(-x)} \) to \( \frac{\sec(x)}{-\tan(x)} \).
This foundational understanding aids in proceeding with further calculations, ultimately leading to successful identity verification.

Secant and Cosecant

The secant and cosecant are reciprocal trigonometric functions derived from the basic sine and cosine functions. They provide a different perspective in equations and are often used in simplifying, transforming, or proving trigonometric identities.

• The secant of an angle \( x \), denoted as \( \sec(x) \), is the reciprocal of the cosine: \( \sec(x) = \frac{1}{\cos(x)} \).
• The cosecant, \( \csc(x) \), is tied to the sine function as its reciprocal: \( \csc(x) = \frac{1}{\sin(x)} \).

In our identity verification, recognizing \( \sec(x) \) and \( \csc(x) \) as reciprocals allows conversion and simplification, transforming \( \sec(x) \) to its reciprocal and connecting it to \( \csc(x) \) through known trigonometric definitions. It forms a basis to express our identity in terms of simpler sine and cosine values of trigonometric expressions.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions that oscillate between -1 and 1. They form the foundation for other trigonometric functions and identities.
The cosine function, \( \cos(x) \), measures the horizontal distance from the origin in a unit circle representation. It is linked to \( \sec(x) \) as \( \sec(x) = \frac{1}{\cos(x)} \).
Sine, expressed as \( \sin(x) \), relates vertical distance and forms the backbone of the tangent function, with \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
These functions are pivotal in expressing the identity. Substituting the definitions into the equation \( -\frac{\sec(x)}{\tan(x)} \) leads us to \( -\frac{1}{\sin(x)} \), revealing the presence of the \( \csc(x) \), ultimately transforming our identity successfully.

Verifying Trigonometric Identities

Verifying trigonometric identities involves transforming one side of the equation to match the other. It requires knowledge of trigonometric properties and an understanding of how to manipulate them.
Initiate with recognition of even and odd functions to simplify expressions. Transition to reciprocals to connect different trigonometric terms like secant and cosecant, rewriting them in terms of basic functions.

• Apply relevant definitions from sine and cosine.
• Simplify expressions step-by-step, like in simplification from \( -\frac{\frac{1}{\cos(x)}}{\frac{\sin(x)}{\cos(x)}} \) to \( -\frac{1}{\sin(x)} \).
• Conclude by equating the identity with known equivalents such as \( -\csc(x) \).

Throughout verification, maintaining systematic steps is key: adjust, rewrite, and simplify progressively to match the right-hand side. The process cultivates deep comprehension and equips students to apply these strategies effectively in solving similar problems.