Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-26: Verify the identity by transforming the lefthand side into the right-hand side. $$ \sin (-x) \sec (-x)=-\tan x $$

Step-by-Step Solution

Answer

The identity is verified as both sides equal \(-\tan x\).

Step 1: Use the Identity for Negative Sine

The first step is to apply the identity for the sine function with a negative angle: \( \sin(-x) = -\sin(x) \). This transforms the expression on the left-hand side from \( \sin(-x) \sec(-x) \) to \( -\sin(x) \sec(-x) \).

Step 2: Use the Identity for Negative Secant

Next, use the identity for the secant of a negative angle: \( \sec(-x) = \sec(x) \). Applying this identity transforms the expression to \( -\sin(x) \sec(x) \).

Step 3: Use the Definition of Secant

Recalling the definition of secant, which is the reciprocal of cosine, we have: \( \sec(x) = \frac{1}{\cos(x)} \). Substituting this into the expression gives \( -\sin(x) \cdot \frac{1}{\cos(x)} \).

Step 4: Simplify Using Tangent Definition

Recognize that the expression \( -\frac{\sin(x)}{\cos(x)} \) is equivalent to \( -\tan(x) \) by the definition of tangent \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Thus, the expression simplifies to \( -\tan(x) \).

Step 5: Conclusion

Now we have turned the left-hand side \( \sin(-x) \sec(-x) \) into \( -\tan(x) \), which matches the right-hand side.

Key Concepts

Negative Angle Identities

Negative angle identities are an important aspect of trigonometry. They describe how trigonometric functions behave when their angle inputs are negative. In simple terms, these identities tell us how to "flip" the angle. For sine and cosine, it's crucial to remember the following identities:

• \( \sin(-x) = -\sin(x) \)
• \( \cos(-x) = \cos(x) \)

These equations suggest that sine is an odd function. This means if you reflect it over the x-axis, it looks the same as multiplying by -1. Cosine, on the other hand, is even. It stays unchanged by reflection over the y-axis. These identities help simplify problems where you have negative angles, turning potentially complex calculations into simpler ones.

Secant

The secant function, denoted by \( \sec(x) \), is the reciprocal of the cosine function. This means \( \sec(x) = \frac{1}{\cos(x)} \). When dealing with negative angles, it’s helpful to use the negative angle identity for secant:

• \( \sec(-x) = \sec(x) \)

This identity shows that secant acts like an even function. It remains the same whether you plug in \( x \) or \( -x \). This property can be especially beneficial when solving trigonometric equations involving negative angles. It simplifies the process since it negates the need to reevaluate the secant function for negative inputs.

Sine

The sine function is fundamental in trigonometry. It measures the "vertical" component of a point on an angle in the unit circle. Sine is characterized as an odd function, meaning that its identity involves a sign change when dealing with negative inputs: \( \sin(-x) = -\sin(x) \). This is important because it allows us to simplify expressions that initially seem complex.

• Odd Function: Reflective over the origin \( (x, y) \to (-x, -y) \)
• Periodicity: Repeats every \(2\pi\)

Understanding sine and its properties such as periodicity help solve many problems in trigonometry, including verifying identities like the one in the original exercise.

Tangent

Tangent is another fundamental trigonometric function, and it is defined as the ratio of sine to cosine: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). This definition is key to understanding tangent's properties:

• Odd Function: Like sine \( \tan(-x) = -\tan(x) \)
• Periodic Function: Repeats every \(\pi\)

The tangent function's identity as an odd function means it behaves similarly to sine concerning negative inputs. In the context of trigonometric identities, tangent serves as a bridge between sine and cosine. By transforming equations like \(-\frac{\sin(x)}{\cos(x)}\) into \(-\tan(x)\), it simplifies solving problems where you must equate expressions together. Understanding these identities allows us to verify many trigonometric equations efficiently.