If \tan^{8}\theta+\cot^{8}\theta=m, then what is the value of \tan\theta+\cot\theta?
- A. \sqrt{\sqrt{m+2}+2}
- B. \sqrt{\sqrt{\sqrt{m+2}+2}+2} ✓
- C. \sqrt{\sqrt{m+2}+2}+2
- D. \sqrt{\sqrt{\sqrt{m+2}+2}+2}+2
Correct Answer: B. \sqrt{\sqrt{\sqrt{m+2}+2}+2}
Explanation
Let x = \tan\theta and 1/x = \cot\theta. Given x^8 + x^{-8} = m. Adding 2 to both sides: (x^4 + x^{-4})^2 = m+2 \implies x^4 + x^{-4} = \sqrt{m+2}. Continuing this process: x^2 + x^{-2} = \sqrt{\sqrt{m+2}+2}. Finally, x + x^{-1} = \sqrt{\sqrt{\sqrt{m+2}+2}+2}.
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