What is the number of solutions of (\sin \theta-\cos \theta)^{2}=2 where -\pi \lt \theta \lt \pi?
- A. <strong>ONLY</strong> one
- B. <strong>ONLY</strong> two ✓
- C. Four
- D. No solution
Correct Answer: B. <strong>ONLY</strong> two
Explanation
Expand the given equation: \sin^2 \theta + \cos^2 \theta - 2\sin \theta \cos \theta = 2. Using identities, 1 - \sin 2\theta = 2 \Rightarrow \sin 2\theta = -1. The given range is -\pi \lt \theta \lt \pi, which means -2\pi \lt 2\theta \lt 2\pi. The solutions for 2\theta in this range are -\frac{\pi}{2} and \frac{3\pi}{2}. Thus, \theta = -\frac{\pi}{4} and \frac{3\pi}{4}. There are exactly two solutions.