Problem 22 - Precalculus: Functions and Graphs

Question

Verify the identity by transforming the left hand side into the right-hand side. $$\csc (-x) \cos (-x)=-\cot x$$

Step-by-Step Solution

Answer

The identity \(\csc(-x) \cos(-x) = -\cot x\) is verified.

Step 1: Use Identities for Negative Angles

Recall the identities for negative angles: \(\csc(-x) = -\csc(x)\) and \(\cos(-x) = \cos(x)\). Substitute these into the left-hand side of the equation:\[ \csc(-x) \cos(-x) = (-\csc(x))(\cos(x)) \]

Step 2: Simplify the Expression

Simplify the expression from Step 1:\[ (-\csc(x))(\cos(x)) = -\csc(x) \cdot \cos(x) \]

Step 3: Use Reciprocal Identities

Recall that \(\csc(x) = \frac{1}{\sin(x)}\). Substitute this identity into the expression:\[ -\csc(x) \cdot \cos(x) = -\left(\frac{1}{\sin(x)}\right) \cdot \cos(x) \]

Step 4: Express in Terms of Cotangent

Simplify the expression from Step 3 using the identity \( \cot(x) = \frac{\cos(x)}{\sin(x)} \):\[ -\left(\frac{1}{\sin(x)}\right) \cdot \cos(x) = -\frac{\cos(x)}{\sin(x)} = -\cot(x) \]

Step 5: Verify the Identity

The transformed left-hand side \(-\cot(x)\) matches the right-hand side of the original identity, thus verifying the identity successfully.

Key Concepts

Negative Angle Identities

Trigonometric functions have specific rules when dealing with negative angles. These rules are known as negative angle identities. They help us understand the behavior of functions when the angle is negative.

• For instance, the cosecant function expresses this as: \( \csc(-x) = -\csc(x) \). This shows that the cosecant of a negative angle is the negative of the cosecant for that angle.
• Similarly, cosine has its identity, where \( \cos(-x) = \cos(x) \). The cosine of an angle remains unchanged even if the angle is negative.

These identities make it easier to manipulate and simplify expressions, especially when verifying trigonometric identities.

Reciprocal Identities

Trigonometric functions have reciprocal relationships, which are very helpful in simplification and solving problems. Understanding these can simplify complex expressions.

• The cosecant \( \csc(x) \) is the reciprocal of sine, expressed as \( \csc(x) = \frac{1}{\sin(x)} \).
• Similarly, secant and tangent also have their reciprocal identities: \( \sec(x) = \frac{1}{\cos(x)} \) and \( \cot(x) = \frac{1}{\tan(x)} \) respectively.

Reciprocals allow us to transform functions into more usable forms, particularly when solving equations or verifying identities.

Cotangent

Cotangent is another basic trigonometric function, defined in terms of sine and cosine. It's essential when dealing with various trigonometric problems.

• The definition of cotangent is \( \cot(x) = \frac{\cos(x)}{\sin(x)} \), making it the ratio of the cosine and sine of an angle.
• Cotangent is often used in problems that involve angles in different quadrants and their properties.

Recognizing this function's relationship to sine and cosine can streamline the process of working with trigonometric identities.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions uses a combination of identities and algebraic manipulation to make them manageable. The goal is often to reduce the expression to a basic or recognizable form.

• Using negative angle identities to replace functions with their equivalent positive angle forms or sign changes.
• Applying reciprocal identities to swap complex trigonometric functions with fractions, easing calculations and simplifications.
• Expressing functions in terms of cotangent or other basic identities can simplify the verification of equations like \( \csc(-x) \cos(-x) = -\cot(x) \).

With practice, these techniques become powerful tools in solving and verifying more complex trigonometric identities.