Consider the following:<br>1. \cos^{4}\theta-\sin^{4}\theta=\frac{2\tan\theta}{1-\tan^{2}\theta}, 0 \lt \theta \lt \frac{\pi}{2}<br>2. \csc\theta+\cot\theta=\frac{1}{\csc\theta-\cot\theta}, 0 \lt \theta \lt \frac{\pi}{2}<br>3. \cos^{2}\theta-\sin^{2}\theta=\frac{1-\tan^{2}\theta}{1+\tan^{2}\theta}, 0 \lt \theta \lt \frac{\pi}{2}<br>Which of the above equations are identities?
- A. 1 and 2 only
- B. 2 and 3 only ✓
- C. 1 and 3 only
- D. 1, 2 and 3
Correct Answer: B. 2 and 3 only
Explanation
Statement 1 LHS is \cos^2\theta - \sin^2\theta = \cos 2\theta, but RHS is \tan 2\theta (False). Statement 2 relies on \csc^2\theta - \cot^2\theta = 1, which is a standard identity, so \csc\theta + \cot\theta = \frac{1}{\csc\theta - \cot\theta} (True). Statement 3 matches the standard double angle formula \cos 2\theta = \frac{1-\tan^2\theta}{1+\tan^2\theta} (True).
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