Problem 90 - Precalculus Essentials

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In order to simplify \(\frac{\cos x}{1-\sin x}-\frac{\sin x}{\cos x},\) I need to know how to subtract rational expressions with unlike denominators.

Step-by-Step Solution

Answer

The statement does not make sense. Trigonometric identities help us to simplify the expression, primarily the Pythagorean Identity, negating the need for subtracting rational expressions with unlike denominators.

Step 1: Assess the Original Expression

Start by taking a look at the original expression \(\frac{\cos x}{1-\sin x}-\frac{\sin x}{\cos x}.\) It clearly is a subtraction of two rational expressions. However, unlike algebraic rational expressions, trigonometric identities can be used to simplify this expression without needing to find a common denominator.

Step 2: Apply Trigonometric Identity

One of the basic identities in trigonometry is the Pythagorean Identity: \(\sin^2 x + \cos^2 x = 1.\) This identity can be rearranged to get either \(\sin x = \sqrt{1 - \cos^2 x}\) or \(\cos x = \sqrt{1 - \sin^2 x},\) both of which can be used to simplify the original expression.

Step 3: Evaluate the Statement

Considering that there's no need for a common denominator due to the trigonometric identities, the statement 'In order to simplify \(\frac{\cos x}{1-\sin x}-\frac{\sin x}{\cos x},\) I need to know how to subtract rational expressions with unlike denominators' is not accurate in this situation.