If sin α, sin β, cos α are in GP, then roots of equation x2 sin β+2x cos β+sin β=0 are
Real
sin α, sin β, cos α in GP⇒sin2β=sin α cos α⇒2 sin2β=2sin α cos α⇒2 sin2β=sin 2α⇒sin2β=12 sin 2x∴ 0≤sin2β≤12 ⇒ cosec2 β≥2
Given equation
x2 sin β+2x cos β+sin β=0⇒x2+2x cot β+1=0∴ Δ=4 cot2β−4=4(cot2 β−1)=4(cosec2β−2)=4(2−2)≥0
∴ real roots