Explanation

Let t = tanθ + cotθ. Then tan²θ + cot²θ = t² − 2, tan⁴θ + cot⁴θ = t⁴ − 4t² + 2, and m = tan⁸θ + cot⁸θ = (t⁴ − 4t² + 2)² − 2. Hence (t² − 2)² = √(m+2) + 2, giving t = √(2 + √(2 + √(m+2))) — the nested-radical option (C). (The option text lost its radical symbols in transcription; as a check, m = 2 gives t = 2.)