In an acute triangle ABC if sides a, b be constants and angle A, B vary then $\frac{\sqrt{{\mathrm{a}}^2-{\mathrm{b}}^2{\sin }^2\mathrm{A}}}{\sqrt{{\mathrm{b}}^2-{\mathrm{a}}^2{\sin }^2\mathrm{B}}}=$ 

We know that $\frac{\mathrm{a}}{\sin \mathrm{A}}=\frac{\mathrm{b}}{\sin \mathrm{B}}\Rightarrow \mathrm{asin}\mathrm{B}=\mathrm{bsin}\mathrm{A}$
$\begin{array}{l}\Rightarrow \mathrm{acos}\mathrm{B\delta B}=\mathrm{bcos}\mathrm{A\delta A}\Rightarrow \frac{\mathrm{\delta A}}{\mathrm{acos}\mathrm{B}}=\frac{\mathrm{\delta B}}{\mathrm{bcos}\mathrm{A}}\\ \Rightarrow \frac{\mathrm{\delta A}}{\sqrt{{\mathrm{a}}^2\left(1-{\sin }^2\mathrm{B}\right)}}=\frac{\mathrm{\delta B}}{\sqrt{{\mathrm{b}}^2\left(1-{\sin }^2\mathrm{A}\right)}}\\ \frac{\mathrm{\delta A}}{\mathrm{\delta B}}=\frac{\sqrt{{\mathrm{a}}^2-a^2{\sin }^2\mathrm{B}}}{\sqrt{{\mathrm{b}}^2-b^2{\sin }^2\mathrm{A}}}\\ \frac{\mathrm{\delta A}}{\mathrm{\delta B}}=\frac{\sqrt{{\mathrm{a}}^2-4R^2{\sin }^{2}A{\sin }^2\mathrm{B}}}{\sqrt{{\mathrm{b}}^2-4R^2{\sin }^{2}B{\sin }^2\mathrm{A}}}\\ \frac{\mathrm{\delta A}}{\mathrm{\delta B}}=\frac{\sqrt{{\mathrm{a}}^2-b^2{\sin }^{2}A}}{\sqrt{{\mathrm{b}}^2-a^2{\sin }^{2}B}}\end{array}$