Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-26: Verify the identity by transforming the lefthand side into the right-hand side. $$ \csc (-x) \cos (-x)=-\cot x $$

Step-by-Step Solution

Answer

The identity is verified.

Step 1: Apply Negative Angle Identities

Recall the trigonometric 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 Expression

Simplify the expression \( (-\csc(x)) \cos(x) \) to get \( -\csc(x) \cos(x) \).

Step 3: Use Reciprocal and Quotient Identities

Recall the identities: \( \csc(x) = \frac{1}{\sin(x)} \) and \( \cot(x) = \frac{\cos(x)}{\sin(x)} \). Substitute \( \csc(x) \) in the equation: \( -\csc(x) \cos(x) = -\left(\frac{1}{\sin(x)}\right) \cos(x) = -\frac{\cos(x)}{\sin(x)} \).

Step 4: Verify Result

Compare \( -\frac{\cos(x)}{\sin(x)} \) to \( -\cot(x) \). Since both expressions are equal, we have successfully transformed the left-hand side into the right-hand side.

Key Concepts

Negative Angle Identities

When dealing with trigonometric functions and their properties, it's essential to comprehend how they behave with negative angles. A negative angle in a geometric context typically suggests rotating in the opposite direction. The identities for negative angles help transform these functions to become more manageable.

• The cosecant of a negative angle is: \( \csc(-x) = -\csc(x) \). This means that the cosecant function is negative when the angle is negative.
• Similarly, the cosine of a negative angle retains its original value: \( \cos(-x) = \cos(x) \). Thus, the cosine function is even, meaning it doesn't change sign with negative angles.

These identities are beneficial because they allow us to simplify and transform trigonometric expressions efficiently. By adjusting these functions, as seen in the exercise, we smoothly transition complex trigonometric equations into simpler forms.

Reciprocal Identities

Reciprocal identities are fundamental tools in trigonometry that express trigonometric functions in terms of their reciprocal relationships. These identities might sound a bit complex at first, but they are straightforward.

• The cosecant is the reciprocal of sine: \( \csc(x) = \frac{1}{\sin(x)} \).
• The secant is the reciprocal of cosine: \( \sec(x) = \frac{1}{\cos(x)} \).
• The cotangent is the reciprocal of tangent: \( \cot(x) = \frac{1}{\tan(x)} \).

In our exercise, we used the fact that \( \csc(x) = \frac{1}{\sin(x)} \) to substitute into the equation. This conversion from cosecant to sine enables further simplification of the given expression, making these reciprocal identities incredibly valuable in proving and verifying identities.

Quotient Identities

Quotient identities describe how tangent and cotangent functions can be represented as a ratio of other trigonometric functions. These identities are another powerful tool in breaking down complex trigonometric expressions.

• The tangent of an angle can be expressed as the ratio of sine over cosine: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
• Conversely, the cotangent can be expressed as the ratio of cosine over sine: \( \cot(x) = \frac{\cos(x)}{\sin(x)} \).

In the exercise provided, we transformed \( -\csc(x) \cos(x) \) into \( -\frac{\cos(x)}{\sin(x)} \), showing it equals \( -\cot(x) \) using these quotient identities. Recognizing when to apply these identities can greatly simplify the verification process in proving trigonometric equations, making them a crucial aspect of solving such problems.