Ifα and βare the roots of the quadratic equation, x2+xsin⁡θ−2sin⁡θ=0,θ∈0,π2,then α12+β12α−12+β−12(α−β)24 isequalto

Given quadratic equation is x2+xsin⁡θ−2sin⁡θ=0,θ∈0,π2and its roots are αand βSo, sum of roots =α+β=−sin⁡θproduct or roots  =αβ=−2sin⁡θ⇒ αβ=2(α+β)……(i)Now, the given express is α12+β12α−12+β−12(α−β)24=α12+β121α12+1β12(α−β)24=α12+β12β12+α12α12β12(α−β)24=αβ(α−β)212=αβ(α+β)2−4αβ12=2(α+β)(α+β)2−8(α+β)12 [ from Eq. (i)] =2(α+β)−812=2−sin⁡θ−812                 [∵α+β=−sin⁡θ]=212(sin⁡θ+8)12