Let ABC be a right angle triangle with ∠C = θ and ∠B = 90°.
By formula,
cosec θ = PerpendicularHypotenuse=ABAC.
Substituting values we get :
⇒15=ABAC
Let AC = 5k and AB = k.
In right angle triangle ABC,
⇒ AC2 = AB2 + BC2
⇒ (5k)2 = BC2 + k2
⇒ 5k2 = BC2 + k2
⇒ BC2 = 5k2 - k2
⇒ BC2 = 4k2
⇒ BC = 4k2
⇒ BC = 2k.
cot θ = PerpendicularBase
= ABBC = k2k = 12
cos θ = HypotenuseBase
= ACBC = 5k2k = 52
∴cot θ - cos θ=12−52=525−2=52(5−1).
Hence, cot θ - cos θ = 52(5−1).