Solve the following equation for 0° < θ ≤ 90°: 3 tan θ + cot θ = 5 cosecθ.

⇒ 3 tan θ + (1/tan θ) = 5 cosec θ

Multiply both sides by tan θ:

⇒ 3tan ² θ + 1 = 5 cosec θ tan θ

⇒ 3tan ² θ + 1 = 5 × (1/sin θ) × (sin θ/cos θ)

⇒ 3(sec ² θ - 1) + 1 = 5 (1/cos θ)

(using 1 + tan ² θ = sec ² θ)

⇒ 3(sec ² θ - 1) + 1 = 5 sec θ

⇒ 3sec ² θ - 5sec θ - 2 = 0

⇒ sec θ =

⇒ sec θ = 2 or (-1/3)
sec θ ≠ (-1/3) as -1 < cos θ <1 )

∴ sec θ = 2

⇒ θ = 60° for 0° < θ ≤ 90°.