Question

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

• A. \( 1 \)
• B. \( -2 \)
• C. \( 2 \)
• D. \( -\frac{3}{2} \)

Hint:

When simplifying complex trigonometric expressions, use standard trigonometric identities to replace terms and simplify the equation step by step.

Answer 1

The Correct Option is C

We are given the expression: \[ \frac{\sin^2 \theta (2 + \cot^2 \theta) - \sin^2 \theta + 2}{\tan^2 \theta + \cot^2 \theta - \sec^2 \theta \csc^2 \theta} \] Let's simplify this expression step by step. 1. Numerator: The numerator is \( \sin^2 \theta (2 + \cot^2 \theta) - \sin^2 \theta + 2 \). We can expand and combine like terms to simplify. 2. Denominator: The denominator is \( \tan^2 \theta + \cot^2 \theta - \sec^2 \theta \csc^2 \theta \). Using trigonometric identities such as \( \tan^2 \theta = \sec^2 \theta - 1 \), \( \cot^2 \theta = \csc^2 \theta - 1 \), and others, we simplify the denominator. After simplification, the entire expression reduces to \( 2 \). Thus, the correct answer is \( 2 \).