In Exercises 67–74, rewrite each expression in terms of the given function or functions. $\frac{1}{1-\cos x}-\frac{\cos x}{1+\cos x}$; $\csc x$

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1. Identify the given expression: \( \frac{\cos x}{1 - \cos x} - \frac{1}{\csc x (1 + \cos x)} \). Our goal is to rewrite this expression in terms of sine and cosine functions, or in terms of the given trigonometric functions.
2. Recall that \( \csc x = \frac{1}{\sin x} \). Substitute this into the expression to rewrite the denominator of the second term: \( \frac{1}{\csc x (1 + \cos x)} = \frac{1}{\frac{1}{\sin x} (1 + \cos x)} = \frac{1}{\frac{1 + \cos x}{\sin x}} \).
3. Simplify the second term by taking the reciprocal of the denominator: \( \frac{1}{\frac{1 + \cos x}{\sin x}} = \frac{\sin x}{1 + \cos x} \). Now the expression becomes \( \frac{\cos x}{1 - \cos x} - \frac{\sin x}{1 + \cos x} \).
4. To combine the two terms, find a common denominator, which is \( (1 - \cos x)(1 + \cos x) \). Recall the Pythagorean identity \( (1 - \cos x)(1 + \cos x) = 1 - \cos^2 x = \sin^2 x \).
5. Rewrite each term with the common denominator \( \sin^2 x \): \( \frac{\cos x (1 + \cos x)}{\sin^2 x} - \frac{\sin x (1 - \cos x)}{\sin^2 x} \). From here, you can combine the numerators over the common denominator and simplify further if needed.

Key Concepts

Reciprocal Trigonometric Functions

Reciprocal functions relate sine, cosine, and tangent to their reciprocals: cosecant (csc), secant (sec), and cotangent (cot). For example, csc x = 1/sin x, which helps rewrite expressions involving csc in terms of sine or cosine.

Pythagorean Identities

Pythagorean identities like sin²x + cos²x = 1 allow conversion between sine and cosine expressions. These identities are essential for simplifying or rewriting trigonometric expressions involving sums or differences of squares.

Algebraic Manipulation of Trigonometric Expressions

Rewriting trigonometric expressions often requires factoring, combining fractions, and rationalizing denominators. Mastery of algebraic techniques enables simplification and expression of complex fractions in terms of a single trigonometric function.