Question

Evaluate \[ \frac{\cosec^2(\theta) - 1}{\cosec^2(\theta)} - \frac{\sec^2(\theta) - 1}{\sec^2(\theta)} \]

• A. \( 2 \cos^2 \theta \)
• B. \( 2 \cos \theta \)
• C. \( 2 \sin^2 \theta \)
• D. \( \cos 2\theta \)
• E. \( 2 \sin \theta \)

Hint:

Use trigonometric identities like \( 1 - \sin^2 \theta = \cos^2 \theta \) to simplify expressions.

Answer 1

The Correct Option is D

We simplify each fraction separately: \[ \frac{\cosec^2(\theta) - 1}{\cosec^2(\theta)} = \frac{\frac{1}{\sin^2 \theta} - 1}{\frac{1}{\sin^2 \theta}} = \frac{\frac{1 - \sin^2 \theta}{\sin^2 \theta}}{\frac{1}{\sin^2 \theta}} = 1 - \sin^2 \theta = \cos^2 \theta \] Similarly, \[ \frac{\sec^2(\theta) - 1}{\sec^2(\theta)} = \frac{\frac{1}{\cos^2 \theta} - 1}{\frac{1}{\cos^2 \theta}} = \frac{\frac{1 - \cos^2 \theta}{\cos^2 \theta}}{\frac{1}{\cos^2 \theta}} = 1 - \cos^2 \theta = \sin^2 \theta \] Now, \[ \frac{\cos^2 \theta}{\sin^2 \theta} = \cot^2 \theta \] Using the identity: \[ \cot^2 \theta = \cos 2\theta \] Thus, the correct answer is \( \cos 2\theta \). 
Final Answer: \[ \boxed{\cos 2\theta} \]