Consider the following:<br><br>1. \frac{\tan \theta+\sin \theta}{\tan \theta-\sin \theta}=\frac{\sec \theta+1}{\sec \theta-1}, where 0 \lt \theta \lt \frac{\pi}{2}<br>2. \frac{\cos^{2}\theta-\sin^{2}\theta}{\cos^{2}\theta+\sin^{2}\theta}=\frac{2 \tan \theta}{\tan^{2}\theta+1}, where 0 \lt \theta \lt \frac{\pi}{2}<br><br>Which of the above is/are identities?
- A. Only 1 ✓
- B. Only 2
- C. Both 1 and 2
- D. Neither 1 nor 2
Correct Answer: A. Only 1
Explanation
Statement 1 simplifies to \frac{\tan\theta+\sin\theta}{\tan\theta-\sin\theta} = \frac{\sin\theta(1/\cos\theta+1)}{\sin\theta(1/\cos\theta-1)} = \frac{\sec\theta+1}{\sec\theta-1}, which is a valid identity. Statement 2 LHS is \cos(2\theta) while RHS is \frac{2\tan\theta}{\sec^2\theta} = \sin(2\theta), making it false.
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