Question

The expression \( \frac{(2\sin^2\theta - 1)(1 + \tan^2\theta)(\sec\theta + \tan\theta)(1 - \sin\theta)}{(\sec^2\theta - 1)(\tan\theta - \cot\theta)\cos^2\theta} \) for \( 0^\circ<\theta<90^\circ \) is equal to:

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

Hint:

To simplify complex trigonometric expressions, use known identities and work through step-by-step simplifications.

Answer 1

The Correct Option is A

Simplify the given expression step by step using trigonometric identities: \[ \text{Use: } \sec^2\theta = 1 + \tan^2\theta, \quad \tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \cos^2\theta = 1 - \sin^2\theta, \quad \csc\theta = \frac{1}{\sin\theta} \] After simplification, you will find that the expression reduces to \( \csc\theta \).