Problem 69 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\csc x-\cot x}{\sec x-1}=\cot x $$

Step-by-Step Solution

Answer

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

Step 1: Express everything in terms of sine and cosine

First, let's recall that \( \csc x = \frac{1}{\sin x} \), \( \cot x = \frac{\cos x}{\sin x} \), and \( \sec x = \frac{1}{\cos x} \). Rewrite the left-hand side (LHS) using these definitions:\[\frac{\csc x - \cot x}{\sec x - 1} = \frac{\frac{1}{\sin x} - \frac{\cos x}{\sin x}}{\frac{1}{\cos x} - 1}\]

Step 2: Simplify the denominator

The denominator \( \frac{1}{\cos x} - 1 \) can be rewritten as a single fraction with a common denominator:\[\frac{1 - \cos x}{\cos x}\]

Step 3: Simplify the numerator

The numerator \( \frac{1}{\sin x} - \frac{\cos x}{\sin x} \) simplifies as:\[\frac{1 - \cos x}{\sin x}\]

Step 4: Combine fractions

Now that both the numerator and denominator are simplified, we find:\[\frac{\frac{1-\cos x}{\sin x}}{\frac{1-\cos x}{\cos x}}\]

Step 5: Simplify the complex fraction

Simplify the complex fraction by multiplying by the reciprocal of the denominator:\[\frac{1-\cos x}{\sin x} \times \frac{\cos x}{1-\cos x} = \frac{\cos x}{\sin x}\]

Step 6: Identify as original expression

The result \( \frac{\cos x}{\sin x} \) is exactly the definition of \( \cot x \), which matches our right-hand side (RHS) of the original equation. Therefore, the identity is verified.

Key Concepts

Reciprocal Identities

In trigonometry, reciprocal identities are fundamental relations between the basic trigonometric functions. They are helpful because they allow us to express one trigonometric function in terms of another using reciprocals. Here's a quick refresher on the reciprocal trigonometric identities:

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

These identities are extremely useful for simplifying trigonometric expressions. If you are given a function in one form, recognizing these identities allows you to switch it into a form that might be easier to work with.

For example, in the original exercise, recognizing that \( \csc x \) and \( \cot x \) can be expressed through \( \sin x \) and \( \cos x \) helps simplify the numerator and makes the whole solution more manageable.

Quotient Identities

Quotient identities are essential when dealing with trigonometric functions, especially when you want to express one function in terms of a quotient of two others. The main ones to remember are:

• The tangent (\( \tan x \)) is represented as:\[ \tan x = \frac{\sin x}{\cos x} \]
• Similarly, the cotangent (\( \cot x \)) is given by:\[ \cot x = \frac{\cos x}{\sin x} \]

These identities are handy for creating fractions that might simplify further or help meet the required expression. In the problem presented, the identity for \( \cot x \) is crucial because that's the expression we want to achieve by manipulating the other side of the equation. Understanding these identities helps you recognize opportunities to transform one side of the identity into another using known relationships between trigonometric functions.

Fraction Simplification

Fraction simplification is a critical skill when working with complex mathematical expressions. It allows us to simplify and combine fractions by employing basic arithmetic and algebraic techniques. Here's how it works in the context of the given problem:1. **Common Denominators**: When you have two fractions that need subtraction or addition, find a common denominator. In the exercise, both the numerator and the denominator were simplified by rewriting them with a single denominator.2. **Complex Fractions**: Often, you have a "fraction of fractions." To simplify these: - Combine the numerator into a single fraction - Combine the denominator into a single fraction3. **Reciprocals to Multiply**: When you divide by a fraction, multiply by its reciprocal. For \[\frac{a/b}{c/d} = \frac{a}{b} \times \frac{d}{c}\].In the solution, the complex fraction was simplified using the reciprocal of the combined denominator. Once common terms are canceled out, you usually end up with a much simpler expression.

In the exercise, understanding how to efficiently simplify fractions using these techniques is essential to verifying that both sides of the equation are indeed the same. This involves recognizing opportunities to cancel terms and to transform fractions into simpler equivalents.