If \tan^2\theta+3 \sec \theta-9=0, where 0 \lt \theta \lt 90^\circ, then what is the value of 12 \cot^2\theta+3 \csc \theta?
- A. (\sqrt{3}+1)^2 ✓
- B. (\sqrt{3}+2)^2
- C. (2\sqrt{3}+1)^2
- D. (3\sqrt{3}+1)^2
Correct Answer: A. (\sqrt{3}+1)^2
Explanation
Substitute \tan^2\theta = \sec^2\theta - 1 to get \sec^2\theta + 3\sec\theta - 10 = 0. Factoring gives (\sec\theta+5)(\sec\theta-2)=0. Thus, \sec\theta = 2 \implies \theta = 60^\circ. Evaluating the expression: 12\cot^2(60^\circ) + 3\csc(60^\circ) = 12(\frac{1}{3}) + 3(\frac{2}{\sqrt{3}}) = 4 + 2\sqrt{3}, which is equal to (\sqrt{3}+1)^2.
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