Problem 150 - Calculus Volume 1

Question

Verify that each equation is an identity. \(\frac{\sin x}{\cos x+1}+\frac{\cos x-1}{\sin x}=0\)

Step-by-Step Solution

Answer

The equation is an identity because both sides equal zero for defined \(x\).

Step 1: Understand the Given Equation

We need to verify if the given equation \( \frac{\sin x}{\cos x+1}+\frac{\cos x-1}{\sin x}=0 \) is an identity. An identity is true for all values of \(x\) where both sides of the equation are defined.

Step 2: Establish Common Denominator

The given equation has two separate fractions. In order to combine them, we need a common denominator. For both fractions, use the product \((\cos x + 1)(\sin x)\) as the common denominator.

Step 3: Rewrite Each Fraction with Common Denominator

Rewrite the fractions with the common denominator:\[\frac{\sin x}{\cos x + 1} = \frac{\sin^2 x}{(\cos x + 1)\sin x}\]\[\frac{\cos x - 1}{\sin x} = \frac{(\cos x - 1)(\cos x + 1)}{(\cos x + 1)\sin x}\]These transformations are valid because we're multiplying the numerator and the denominator by the same terms.

Step 4: Simplify Each Fraction

Now expand and simplify each fraction:- The first term becomes \( \frac{\sin^2 x}{(\cos x + 1)\sin x} \).- For the second term, use the identity \((\cos x - 1)(\cos x + 1) = \cos^2 x - 1^2 = \cos^2 x - 1\). This becomes \( \frac{\cos^2 x - 1}{(\cos x + 1)\sin x} \).

Step 5: Combine and Simplify

Combine the fractions:\[\frac{\sin^2 x}{(\cos x + 1)\sin x} + \frac{\cos^2 x - 1}{(\cos x + 1)\sin x} = \frac{\sin^2 x + \cos^2 x - 1}{(\cos x + 1)\sin x} \]Use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) which simplifies the numerator: \[1 - 1 = 0\]

Step 6: Verify Result

Since the numerator becomes zero, the entire expression simplifies to:\[\frac{0}{(\cos x + 1)\sin x} = 0\]Thus, the original equation is an identity as it holds true for any \(x\) where the functions are defined.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry. It states that for any angle \(x\), the square of the sine of \(x\) plus the square of the cosine of \(x\) will always equal 1. This is expressed mathematically as:

• \( \sin^2 x + \cos^2 x = 1 \)

This identity is derived from the Pythagorean Theorem and forms the basis for many trigonometric transformations and simplifications.
In this exercise, we used the Pythagorean Identity to simplify the equation:

• Replace \(\sin^2 x + \cos^2 x\) in the numerator with 1.
• The rest of the expression can then be simplified further.

Understanding this identity allows you to see why the trigonometric equation simplifies to 0, confirming it as an identity.

Common Denominator

A common denominator is crucial when adding or subtracting fractions. It allows multiple fractions to be combined into a single expression.
In our problem, we have two fractions with different denominators:

• \( \frac{\sin x}{\cos x + 1} \)
• \( \frac{\cos x - 1}{\sin x} \)

To add these fractions, we need a common denominator. A suitable choice is the product \((\cos x + 1)(\sin x)\).
This method helps because it covers both original denominators and allows the fractions to be:

• Rewritten with the same denominator \((\cos x + 1)\sin x\).
• Easily combined into a single fraction.

Once this is done, we can proceed with simplifying the entire expression further.

Fractions Simplification

Simplifying fractions involves reducing fractions to their simplest form. This often includes:

• Multiplying or dividing the numerator and the denominator by the same value.
• Using identities or properties to simplify terms.

In the given exercise, we took specific steps to simplify our expression:

• Multiply and simplify: Each fraction is adjusted to have the common denominator, making it possible to combine them.
• Simplification with identities: By invoking the identity \((\cos x - 1)(\cos x + 1) = \cos^2 x - 1\), further letting us simplify.
• Use of the Pythagorean identity: After combining, simplify \(\sin^2 x + \cos^2 x - 1\) to 0, since \(\sin^2 x + \cos^2 x = 1\).

Combining all these techniques allows you to effectively reduce complex expressions into simpler, verifiable forms. It's essential to systematically apply algebraic rules and trigonometric identities during this process.