Problem 23 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The identity is verified as \( \frac{\cot (-x)}{\csc (-x)} = \cos(x) \).

Step 1: Simplify the Cotangent

We start by simplifying the left side of the equation. The cotangent function, \( \cot(-x) \), can be rewritten using the cofunction identity as \( -\cot(x) \). Hence, \( \frac{\cot (-x)}{\csc (-x)} \) becomes \( \frac{-\cot (x)}{\csc (-x)} \).

Step 2: Simplify the Cosecant

Next, we simplify the cosecant. The function \( \csc(-x) \) can be expressed using the identity \( \csc(-x) = -\csc(x) \), because cosecant is an odd function. This means our expression is now \( \frac{-\cot(x)}{-\csc(x)} \).

Step 3: Cancel the Negative Signs

In both the numerator and the denominator of our expression \( \frac{-\cot(x)}{-\csc(x)} \), we have negative signs that can be cancelled out. This simplifies the expression to \( \frac{\cot(x)}{\csc(x)} \).

Step 4: Substitute Cotangent and Cosecant

We replace \( \cot(x) \) with \( \frac{\cos(x)}{\sin(x)} \) (since \( \cot(x) = \frac{\cos(x)}{\sin(x)} \)) and \( \csc(x) \) with \( \frac{1}{\sin(x)} \) (as \( \csc(x) = \frac{1}{\sin(x)} \)). Thus, the expression becomes: \( \frac{\frac{\cos(x)}{\sin(x)}}{\frac{1}{\sin(x)}} \).

Step 5: Simplify the Fraction

To simplify, solve the complex fraction \( \frac{\frac{\cos(x)}{\sin(x)}}{\frac{1}{\sin(x)}} \) by multiplying the numerator \( \frac{\cos(x)}{\sin(x)} \) by the reciprocal of the denominator \( \sin(x) \). This gives: \( \frac{\cos(x)}{\sin(x)} \times \sin(x) = \cos(x) \).

Step 6: Conclude the Identity

Since, through simplification, the left side \( \frac{\cot (-x)}{\csc (-x)} \) equals \( \cos(x) \), we have verified that the initial equation is indeed an identity.

Key Concepts

Cofunction Identities

Trigonometric cofunction identities link angles that complement to 90 degrees (or \(\pi/2\) radians). A key aspect of these identities is that the value of a trigonometric function of an angle is equal to the cofunction of its complement. In simple terms:

• \( \sin(90^\circ - x) = \cos(x) \)
• \( \cos(90^\circ - x) = \sin(x) \)
• \( \tan(90^\circ - x) = \cot(x) \)
• \( \cot(90^\circ - x) = \tan(x) \)
• \( \sec(90^\circ - x) = \csc(x) \)
• \( \csc(90^\circ - x) = \sec(x) \)

If you consider the cofunction identity for cotangent \( \cot(-x) \), it turns into \( -\cot(x) \). This simplification shows how cofunction identities can transform trigonometric functions when dealing with complementary angles. Understanding this concept is essential, as it serves as a foundational piece when solving identities by transforming one side into another.

Odd and Even Functions

Odd and even functions are a fundamental concept in mathematics, especially within trigonometry. Functions are categorized based on their symmetry:

• Odd Functions: A function \( f(x) \) is considered odd if \( f(-x) = -f(x) \). Graphically, odd functions are symmetric about the origin.
• Even Functions: A function \( f(x) \) is even if \( f(-x) = f(x) \). These functions are symmetric about the y-axis.

In trigonometry:

• The sine (\( \sin(x) \)) and cosine (\( \cos(x) \)) functions are even, meaning their values remain unchanged if you reflect across the y-axis.
• The tangent (\( \tan(x) \)), cotangent (\( \cot(x) \)), secant (\( \sec(x) \)), and cosecant (\( \csc(x) \)) functions are odd, implying they change sign when the angle is negated.

Understanding these properties is crucial as they guide the simplification of trigonometric expressions, such as the provided solution where \( \csc(-x) = -\csc(x) \). This inherent symmetry helps in verifying identities and simplifying equations.

Complex Fractions Simplification

Simplifying complex fractions involves reducing expressions that have fractions both in their numerator and denominator. It is essential to simplify them to solve and understand equations efficiently. Here's how you can simplify complex fractions:

• Identify the Fractions: Look for fractions in both the numerator and the denominator. In the case of trigonometric functions, these might be expressions like \( \frac{\cos(x)}{\sin(x)} \) (cotangent) or \( \frac{1}{\sin(x)} \) (cosecant).
• Combine the Expression: In a complex fraction, rewrite the fraction as something more manageable. This often means combining or multiplying by the reciprocal.
• Cancel Out Units: You can simplify by cancelling out terms that appear in both the numerator and the denominator.

For instance, in the given solution, the complex fraction \( \frac{\frac{\cos(x)}{\sin(x)}}{\frac{1}{\sin(x)}} \) simplifies by multiplying by the reciprocal to become \( \cos(x) \). This process is especially useful in trigonometry to verify identities by breaking down complex expressions into their simplest form.