Simplifying the Trigonometric Expression

The goal is to simplify the given expression: $ \frac{\cot\theta - \cos\theta}{\cot\theta + \cos\theta} $ We will use trigonometric identities to simplify this expression.

Step 1: Express Cotangent in Terms of Sine and Cosine

Recall the identity $\cot\theta = \frac{\cos\theta}{\sin\theta}$. Substitute this into the expression:

$ \frac{\frac{\cos\theta}{\sin\theta} - \cos\theta}{\frac{\cos\theta}{\sin\theta} + \cos\theta} $

Step 2: Factor out Cosine

Factor out the common term $\cos\theta$ from both the numerator and the denominator:

$ \frac{\cos\theta \left( \frac{1}{\sin\theta} - 1 \right)}{\cos\theta \left( \frac{1}{\sin\theta} + 1 \right)} $

Step 3: Cancel Common Factors

Cancel the $\cos\theta$ term from the numerator and denominator, assuming $\cos\theta \neq 0$:

$ \frac{\frac{1}{\sin\theta} - 1}{\frac{1}{\sin\theta} + 1} $

Step 4: Use the Cosecant Identity

Use the identity $\csc\theta = \frac{1}{\sin\theta}$. Substitute this into the simplified expression:

$ \frac{\csc\theta - 1}{\csc\theta + 1} $ This simplified form matches option B.