Consider the following statements:<br>1. The equation 2 \sin^{2}\theta - \cos \theta + 4 = 0 is possible for <strong>ALL</strong> \theta.<br>2. \tan \theta + \cot \theta <strong>CANNOT</strong> be less than 2, where 0 \lt \theta \lt \frac{\pi}{2}.<br>Which of the above statements is/are correct?

Explanation

For statement 1: 2(1-\cos^2\theta) - \cos\theta + 4 = 0 \Rightarrow 2\cos^2\theta + \cos\theta - 6 = 0. Solving this gives \cos\theta = \frac{3}{2} or -2, which are outside the valid range [-1, 1]. Hence, it is not possible. For statement 2: Using AM \geq GM, \tan\theta + \frac{1}{\tan\theta} \geq 2\sqrt{\tan\theta \times \frac{1}{\tan\theta}} = 2 for all positive values. Thus, statement 2 is true.

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