Problem 68 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\frac{1-\cos x}{\sin x}+\frac{\sin x}{1-\cos x}=2 \csc x$$

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \( 2 \csc x \).

Step 1: Understand the Expression

In this identity verification problem, we need to show \[ \frac{1-\cos x}{\sin x} + \frac{\sin x}{1-\cos x} = 2 \csc x. \]First, let's identify a strategy to simplify both sides.

Step 2: Simplify the Left Side

Start with the left side of the identity. It is\[ \frac{1-\cos x}{\sin x} + \frac{\sin x}{1-\cos x}. \]To simplify, find a common denominator which is \( \sin x(1-\cos x) \).This gives:\[ \frac{(1-\cos x)^2 + \sin^2 x}{\sin x (1-\cos x)}. \]

Step 3: Expand and Simplify the Numerator

Expand \((1-\cos x)^2 = 1 - 2\cos x + \cos^2 x\).So the new expression for the numerator becomes:\[ 1 - 2\cos x + \cos^2 x + \sin^2 x. \]

Step 4: Apply Pythagorean Identity

Use the identity \( \sin^2 x + \cos^2 x = 1 \) to further simplify the numerator:\[ 1 - 2\cos x + \cos^2 x + \sin^2 x = 2 - 2\cos x. \]

Step 5: Factor Out a Constant

From the previous step, the expression becomes:\[ \frac{2(1 - \cos x)}{\sin x (1-\cos x)}. \]This simplifies to:\[ \frac{2}{\sin x}. \]

Step 6: Express in Terms of Cosecant

Recognize that \( \frac{2}{\sin x} \) is equivalent to:\[ 2 \csc x. \]

Step 7: Compare with the Right Side

The right side of the equation is already \( 2 \csc x \).Since both sides are equal, the identity is verified.

Key Concepts

Pythagorean identity

The Pythagorean identity is a cornerstone of trigonometry and is crucial for simplifying expressions involving trigonometric functions. It is expressed as \[ \sin^2 x + \cos^2 x = 1. \]This identity arises from the Pythagorean Theorem applied to a right triangle on the unit circle. Understanding this relationship highlights how the squares of the sine and cosine of any angle add up to 1.

In the context of the given identity exercise, the Pythagorean identity helps simplify the complex expression. Particularly, the expression \( 1 - 2\cos x + \cos^2 x + \sin^2 x \)can be rewritten using \( \sin^2 x + \cos^2 x = 1 \).This single substitution greatly reduces complexity, showing how powerful this identity can be when working with trigonometric formulas. Whenever tackling a trigonometric problem, keep the Pythagorean identity in mind, as it often serves as a key to untangling complicated expressions.

cosecant function

The cosecant function, indicated as \( \csc x \),is the reciprocal of the sine function. It is defined as \( \csc x = \frac{1}{\sin x} \).This function is especially useful when dealing with reciprocal identities in trigonometry.

In the verification of the trigonometric identity \( \frac{1-\cos x}{\sin x} + \frac{\sin x}{1-\cos x} = 2 \csc x \),the goal is to express the simplified left side in terms of cosecant. The step-by-step solution elegantly transforms the left side of the equation to \( \frac{2}{\sin x} \),which is precisely \( 2 \csc x \).

Recognizing expressions that can be rewritten using the cosecant function is vital for simplifying identities and verifying equalities in trigonometric problems. Whenever you see a \( \frac{1}{\sin x} \)in any form, think \( \csc x \).

common denominator

Finding a common denominator is a fundamental technique in mathematics, essential for combining fractions. It means finding a shared base under the fraction bars so that the fractions can be added or subtracted.

In the problem \( \frac{1-\cos x}{\sin x} + \frac{\sin x}{1-\cos x} \),finding a common denominator simplifies the process immensely. The chosen common denominator here is \( \sin x (1-\cos x) \).

Once both fractions are expressed with this common base, their numerators can be summed directly: \( \frac{(1-\cos x)^2 + \sin^2 x}{\sin x (1-\cos x)} \).Without a common denominator, combining fractions would be complicated and error-prone, especially with trigonometric expressions. Ensuring that you properly manage these denominators will lead to clearer and more manageable solutions. Keeping your fractions tidy helps in maintaining an organized approach throughout your verification process.