Problem 81 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \(\frac{1-\cos x}{1+\cos x}=(\cot x-\csc x)^{2}\)

Step-by-Step Solution

Answer

The given equation is indeed an identity since both sides simplify to \( \frac{1 - \cos x}{1 + \cos x} \).

Step 1: Understand the Identity

We need to verify that the equation \( \frac{1-\cos x}{1+\cos x} = (\cot x - \csc x)^2 \) is an identity. This means these two expressions must be equal for all values of \( x \) for which both sides are defined.

Step 2: Simplify the Left Side

The left side is \( \frac{1-\cos x}{1+\cos x} \). There is no obvious way to simplify this directly; usually, it is compared or simplified using a trigonometric identity, but let's move to simplify the right side first.

Step 3: Simplify the Right Side

The right side is \( (\cot x - \csc x)^2 \). Since \( \cot x = \frac{\cos x}{\sin x} \) and \( \csc x = \frac{1}{\sin x} \), substitute these into the expression: \[(\frac{\cos x}{\sin x} - \frac{1}{\sin x})^2.\] Simplify inside the parentheses to get: \[(\frac{\cos x - 1}{\sin x})^2.\] When squared, it becomes: \[\frac{(\cos x - 1)^2}{\sin^2 x}.\] Use the Pythagorean identity \( \sin^2 x = 1 - \cos^2 x \) to rewrite: \[\frac{(1 - \cos x)^2}{1 - \cos^2 x}.\] This is equivalently \( \frac{(1 - \cos x)^2}{(1 + \cos x)(1 - \cos x)} \). Cancel a common factor \((1 - \cos x)\) from numerator and denominator to get: \[\frac{1 - \cos x}{1 + \cos x}.\]

Step 4: Compare and Conclude

The simplified right side \( \frac{1 - \cos x}{1 + \cos x} \) matches exactly the left side as derived. Thus, \( \frac{1 - \cos x}{1 + \cos x} = (\cot x - \csc x)^2 \) is verified as an identity.

Key Concepts

Verification of Identities

In trigonometry, verifying identities involves confirming that two different expressions are equal for all values within their domain. This is crucial in simplifying expressions and solving equations. Here, we aim to verify that the equation \( \frac{1-\cos x}{1+\cos x}=(\cot x-\csc x)^{2} \) is indeed an identity.

To verify, you need to show both sides are equivalent by manipulating at least one side of the equation using known identities or simplifications.
Key tasks in this process include:

• Identifying possible identities or rules that might apply.
• Simplifying one or both expressions to see if they equate.
• Using logical steps to show that each manipulation retains equality.

For this exercise, the verification process begins by transforming the right side into the same form as the left. Confirming that both resemble each other after simplification completes the verification process effectively.

Trigonometric Simplification

Trigonometric simplification is a technique used to make trig expressions more manageable and to reveal hidden equalities. In our exercise, the expression \((\cot x - \csc x)^2\) requires simplification to match the left side \(\frac{1-\cos x}{1+\cos x}\).

This simplification process involves breaking down complex expressions into simpler trigonometric functions. Here's how it works:

• Recognize basic trigonometric identities, such as \( \cot x = \frac{\cos x}{\sin x} \) and \( \csc x = \frac{1}{\sin x} \).
• Substitute these basic identities into the expression.
• Simplify fractions and like terms, aiming to collapse complex terms into familiar forms.

In our case, we simplified \((\frac{\cos x}{\sin x} - \frac{1}{\sin x})^2\) which ultimately reduced to the same form as our initial left side expression.

Pythagorean Identity

The Pythagorean identity is a pivotal element in trigonometry, relating the primary trigonometric functions with an equation like \( \sin^2 x + \cos^2 x = 1 \). This identity serves not only as a creed to check correctness but also facilitates simplification.

During the exercise, the Pythagorean identity helps in transitioning expressions:

• Notice that \( \sin^2 x \) can be expressed as \( 1 - \cos^2 x \).
• Use this form to reframe the expression and help cancel out terms.
• Integrate the identity to simplify expressions systematically into equivalent forms.

Within the solution process, we employed the identity to rearrange and simplify the right side, achieving meaningful cancellation which confirmed both sides of the identity are the same. Using Pythagorean identity is an invaluable strategy in solving such trigonometric challenges.