3tan θ + cot θ = 5 cosec θ. Solve for θ,0 <θ≤ 90.
3sinθcosθ + cosθsinθ = 5sinθ
sinθ ( 3sin2θ + cos2θ) = 5 sinθ cosθ
3(1 - cos2θ) + cos2θ = 5 cosθ
2cos2θ + 5cosθ - 3 = 0
2cosθ [cosθ + 3] - 1(cosθ + 3) = 0
(cosθ + 3) (2cosθ - 1) = 0
cosθ = -3 or cosθ = 12
cosθ = -3 not possible.
Hence θ = 60∘ from cosθ = 12