Explanation

Note: the question clause is truncated; with options √3/4, 1/√3, √3, 4/√3 it asks for tanθ+cotθ (equivalently secθ·cosecθ). From (sinθ+cosθ)² = 1+2sinθcosθ = (1+√3)²/4 we get sinθcosθ = √3/4, so tanθ+cotθ = 1/(sinθcosθ) = 4/√3 (option D).