Problem 71 - Precalculus Essentials

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

\(\frac{1}{\csc x} - \sqrt{1-\frac{1}{\csc ^2 x}}\)

Step 1: Understand the Trigonometric Identity

The key trigonometric identity to use in this situation is \( \csc^2 x = 1 + \cot^2 x \). This identity allows to rewrite any expression involving cosecant squared in terms of cotangent squared or vice versa.

Step 2: Express Given Function in terms of Sine and Cosine

To start off, express each part of the expression in terms of sine and cosine. As \( \csc x = \frac{1}{\sin x} \) - this will be helpful later on. So the expression becomes \( \frac{1}{1-\cos x}-\frac{\cos x}{1+\cos x} = \frac{\sin^2 x}{\sin x(1-\cos x)}-\frac{\sin x \cos x}{\sin x(1+\cos x)} \)

Step 3: Simplifying the expression

The resulting expression can now be combined over a common denominator and simplified. It reduces to \( \frac{\sin^2 x - \sin x \cos x}{\sin x} = \frac{\sin x (\sin x - \cos x)}{\sin x} = \sin x - \cos x \). This can be further simplified using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) to rewrite the \( \cos x \) term as \( \sin x - \sqrt{(1-\sin^2 x)} \)

Step 4: Rewrite in terms of cosecant

Finally, substitute \( \sin x \) with \( \frac{1}{\csc x} \). After substitution, the expression becomes \( \frac{1}{\csc x} - \sqrt{1-\frac{1}{\csc ^2 x}} \). This is the final expression in terms of \( \csc x \).