Question

The value of \( \frac{(1 + \cot \theta - \csc \theta)(1 + \tan \theta + \sec \theta)}{\tan^2 \theta + \cot^2 \theta - \sec^2 \theta \csc^2 \theta} \) is:

• A. -2
• B. -1
• C. 1
• D. 2

Hint:

When dealing with complex trigonometric expressions, use fundamental identities to simplify terms step by step.

Answer 1

The Correct Option is B

We are tasked with simplifying the given expression: \[ \frac{(1 + \cot \theta - \csc \theta)(1 + \tan \theta + \sec \theta)}{\tan^2 \theta + \cot^2 \theta - \sec^2 \theta \csc^2 \theta} \] First, apply the standard trigonometric identities to break down the terms: \[ \cot \theta = \frac{\cos \theta}{\sin \theta}, \quad \csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \tan \theta = \frac{\sin \theta}{\cos \theta} \] Substituting these into the expression: \[ \text{Numerator: } (1 + \frac{\cos \theta}{\sin \theta} - \frac{1}{\sin \theta})(1 + \frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta}) \] Simplifying each term: \[ \text{Numerator: } \left( \frac{\sin \theta + \cos \theta - 1}{\sin \theta} \right) \left( \frac{\cos \theta + \sin \theta + 1}{\cos \theta} \right) \] Now, simplify the denominator: \[ \tan^2 \theta + \cot^2 \theta - \sec^2 \theta \csc^2 \theta \] Using the identity \( \tan^2 \theta + \cot^2 \theta = \sec^2 \theta \csc^2 \theta - 1 \), the denominator simplifies to: \[ 1 \] Thus, the whole expression simplifies to: \[ (1 + \cot \theta - \csc \theta)(1 + \tan \theta + \sec \theta) = -1 \] Thus, the value of the given expression is \( -1 \).