Question

The expression \( \frac{(1 + \tan \theta) \cos \theta}{\sin \theta \tan \theta (1 - \tan \theta) + \sin \theta \sec^2 \theta} \) is equal to:

• A. \( \sec \theta \)
• B. \( \csc \theta \)
• C. \( \cot \theta \)
• D. \( \tan \theta \)

Hint:

To simplify trigonometric expressions, always look for opportunities to factor out common terms, and use known trigonometric identities like \( \sec^2 \theta = 1 + \tan^2 \theta \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Answer 1

The Correct Option is A

We are given the expression: \[ \frac{(1 + \tan \theta) \cos \theta}{\sin \theta \tan \theta (1 - \tan \theta) + \sin \theta \sec^2 \theta} \] Let's simplify the expression step by step. 1. Start by factoring out \( \sin \theta \) from the denominator: \[ \sin \theta \left( \tan \theta (1 - \tan \theta) + \sec^2 \theta \right) \] 2. We know that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), and substitute this into the equation: \[ \sin \theta \left( \frac{\sin \theta}{\cos \theta} (1 - \frac{\sin \theta}{\cos \theta}) + \frac{1}{\cos^2 \theta} \right) \] 3. After simplifying the terms, we find that the expression simplifies to \( \sec \theta \), which is the correct answer. Thus, the correct answer is \( \sec \theta \).