If 14\sin^{2}\theta+10\cos^{2}\theta=11 where 0^{\circ}\lt\theta\lt90^{\circ}, then what is the value of \tan\theta+\cot\theta?
- A. \frac{4}{\sqrt{3}} ✓
- B. \frac{2}{\sqrt{3}}
- C. \sqrt{3}
- D. 2\sqrt{3}
Correct Answer: A. \frac{4}{\sqrt{3}}
Explanation
Substituting \cos^2\theta = 1 - \sin^2\theta, we get 14\sin^2\theta + 10 - 10\sin^2\theta = 11 \implies 4\sin^2\theta = 1 \implies \sin\theta = \frac{1}{2}. Thus, \theta = 30^{\circ}. Then, \tan 30^{\circ} + \cot 30^{\circ} = \frac{1}{\sqrt{3}} + \sqrt{3} = \frac{4}{\sqrt{3}}.
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