Explanation

Let t = 64^(sin²θ); then 64^(cos²θ) = 64^(1−sin²θ) = 64/t. The equation t + 64/t = 16 gives t² − 16t + 64 = (t−8)² = 0, so t = 8 = 64^(1/2). Thus sin²θ = 1/2, and with 0 ≤ θ ≤ π/2, θ = 45°. Then tanθ + cotθ = 1 + 1 = 2 — option B.