Problem 71 - Algebra and Trigonometry

Question

Rewrite each expression in terms of the given function or functions. \(\frac{1}{1-\cos x}-\frac{\cos x}{1+\cos x} ; \csc x\)

Step-by-Step Solution

Answer

The expression \( \frac{1}{1 - \cos x} - \frac{\cos x}{1 + \cos x} \) rewritten in terms of cicant function \( \csc x \) is \( \csc^{2}x - \csc x \sec x \)

Step 1: Utilize Pythagorean Identity

Use the identity \(1 - \cos^2 x = \sin^2 x\) to replace the \(1 - \cos x\) and \(1 + \cos x\) in the denominator. Remember that \(\csc x\) is the reciprocal of \(\sin x\). Thus, the expression is rewritten as: \(\frac{1}{\sin^2x} - \frac{\cos x}{\sin^{2}x} \)

Step 2:Factor Out \( \csc^2 x \)

The expressions in the denominators can be converted to \( \csc^2 x \) since it's the reciprocal. The expression will then become: \( \csc^{2}x(1 - \cos x) \)

Step 3: Simplify the Expression

Using the reciprocal of cosine function which is secant function (\(\sec x \)), the expression can be transformed to: \( \csc^{2}x(1 - \sec x) \). As a final simplification, this can be rewritten as: \( \csc^{2}x - \csc x \sec x \).

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the fundamental identities in trigonometry. It relates the square of the sine function to the square of the cosine function. The identity states that:

• \( \sin^2 x + \cos^2 x = 1 \)

This can be rearranged to solve for either \( \sin^2 x \) or \( \cos^2 x \), depending on the problem at hand. In our exercise, we used the identity to rewrite \( 1 - \cos^2 x \) as \( \sin^2 x \).
This transformation is crucial for simplifying expressions involving trigonometric functions, especially when dealing with complex fractions.

• Remember, substituting \( \sin^2 x \) or \( \cos^2 x \) allows you to work within a simpler trigonometric framework, often leading to a more straightforward solution.

Understanding the Pythagorean identity helps in manipulating and simplifying various trigonometric expressions.

Cosecant Function

The cosecant function, denoted as \( \csc x \), is the reciprocal of the sine function. It is defined as:

• \( \csc x = \frac{1}{\sin x} \)

This reciprocal relationship means that whenever you see \( \csc x \), you can think of it in terms of \( \sin x \). For example, if \( \sin x = a \), then \( \csc x = \frac{1}{a} \).
In the exercise, recognizing the relationship between \( \csc x \) and \( \sin x \) allowed us to express the terms with \( \sin^2 x \) in terms of \( \csc^2 x \).

• This simplification was key in expressing the entire original expression in a form that is easier to work with.

Understanding the cosecant function and being comfortable with its reciprocal nature in relation to sine will greatly aid in solving trigonometric problems that involve these transformations.

Reciprocal Identities

Reciprocal identities in trigonometry are a set of identities that express each basic trigonometric function as the reciprocal of another trigonometric function. The key reciprocal identities are:

• \( \csc x = \frac{1}{\sin x} \)
• \( \sec x = \frac{1}{\cos x} \)
• \( \cot x = \frac{1}{\tan x} \)

These identities are incredibly useful in trigonometry because they allow you to switch between functions, simplifying expressions or solving equations.
For example, converting \( \frac{1}{\sin^2 x} \) directly to \( \csc^2 x \) simplifies the exercise presented in a more manageable form. Similarly, recognizing \( \sec x \) as \( \frac{1}{\cos x} \) enables transformation into expressions that are easier to interpret.

• The use of reciprocal identities allows for more flexibility and efficiency in simplifying trigonometric expressions, and is important for solving a wide variety of trigonometry problems.

By mastering these identities, you'll find it easier to work with more complex expressions and equations in trigonometry.