Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The identity is verified, left-hand side simplifies to \(-\csc x\).

Step 1: Simplify the Left-Hand Side

To verify the identity \( \frac{\sec (-x)}{\tan (-x)}=-\csc x \), we start by simplifying the left-hand side. First, recall the identities for secant and tangent of a negative angle: \( \sec(-x) = \sec(x) \) and \( \tan(-x) = -\tan(x) \). Substituting these into the expression, we get \( \frac{\sec(x)}{-\tan(x)} \).

Step 2: Simplify the Fraction

The expression \( \frac{\sec(x)}{-\tan(x)} \) can be rewritten as \( -\frac{\sec(x)}{\tan(x)} \). This simplifies our process by factoring out the negative sign.

Step 3: Rewrite in Terms of Sine and Cosine

Now express \( \sec(x) \) and \( \tan(x) \) in terms of sine and cosine: \( \sec(x) = \frac{1}{\cos(x)} \) and \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Thus, \( \frac{\sec(x)}{\tan(x)} = \frac{\frac{1}{\cos(x)}}{\frac{\sin(x)}{\cos(x)}} \).

Step 4: Simplify the Complex Fraction

Simplify the complex fraction \( \frac{\frac{1}{\cos(x)}}{\frac{\sin(x)}{\cos(x)}} \) by multiplying both the numerator and the denominator by \( \cos(x) \). This gives \( \frac{1}{\sin(x)} \).

Step 5: Recognize the Definition of Cosecant

Recognize that \( \frac{1}{\sin(x)} = \csc(x) \), thus making \( \frac{\sec(x)}{\tan(x)} = \csc(x) \). Including the negative sign from Step 2, we have \( -\csc(x) \).

Step 6: Conclude the Verification

The left-hand side simplifies to \( -\csc(x) \), which matches the right-hand side of the original equation, thus verifying the identity.

Key Concepts

Understanding Secant

Secant, often abbreviated as "sec," is a trigonometric function that relates to the cosine function. It is the reciprocal of the cosine function, meaning it simply is the inverse value of cosine. In mathematical terms, secant can be expressed as: \[ \sec(x) = \frac{1}{\cos(x)} \] This relationship implies that wherever cosine is non-zero, secant is well-defined.

• When the angle \( x \) is such that \( \cos(x) = 0 \), the secant function is undefined.
• The secant function has vertical asymptotes wherever the cosine function is zero, primarily at odd multiples of \( \frac{\pi}{2} \).

In the context of verifying trigonometric identities, secant helps transform expressions into more manageable forms, especially when combined with other functions like tangent and sine.

Exploring Tangent

Tangent, or "tan", is one of the primary trigonometric functions. It is defined as the ratio of the sine and cosine of an angle: \[ \tan(x) = \frac{\sin(x)}{\cos(x)} \] This means that tangent is essentially a measure of the ratio between the opposite and adjacent sides of a right triangle.

• An important property of tangent is that it becomes undefined when \( \cos(x) = 0 \), leading to vertical asymptotes on its graph.
• The tangent function is periodic with a period of \( \pi \), meaning that its pattern repeats every \( \pi \, \, ext{radians} \) or \( 180^{\circ} \).

In trigonometric identities, tangent frequently appears due to its relationships with sine and cosine, facilitating simplifications and transformations.

Defining Cosecant

Cosecant, abbreviated as "csc", is another trigonometric function that serves as the reciprocal of sine. It is given by the expression: \[ \csc(x) = \frac{1}{\sin(x)} \] This implies cosecant is defined whenever the sine of the angle is not zero.

• Wherever \( \sin(x) = 0 \), the cosecant function is undefined.
• The graph of the cosecant function has vertical asymptotes wherever the sine function is zero, at integer multiples of \( \pi \).

Cosecant is commonly used in identity verifications to simplify complex fractions and connect different trigonometric functions through these reciprocal relationships.