Preserving Sines

Relevant wiki: Sum and Difference Trigonometric Formulas - Problem Solving

Let $y=\sqrt{9-8\sin50^\circ}$

Hint in problem : mention of triple angle formula as well as " $\text{csc}$ " which prompt to multiply numerator and denominator by $\sin(x)$

So, $y=\frac{\sqrt{9\sin^2(50^\circ)-8\sin^3(50^\circ)}}{\sin(50^\circ)}$ $=\frac{\sqrt{9\sin^2(50^\circ)-6\sin(50^\circ)+6\sin(50^\circ)-8\sin^3(50^\circ)}}{\sin(50^\circ)}$ $=\frac{\sqrt{9\sin^2(50^\circ)-6\sin(50^\circ)+2\times \sin(3\times 50^\circ)}}{\sin(50^\circ)}$ $=\frac{\sqrt{9\sin^2(50^\circ)-6\sin(50^\circ)+1}}{\sin(50^\circ)}$ $=\frac{|3\sin(50^\circ)-1|}{\sin(50^\circ)}$ $\Rightarrow a\times b=\boxed{-3}$