Simplify this expression

We have 
\[\frac{\sin^{-2}x+\cos^{-2}x}{(\sin^{-1}x+\cos^{-1}x)^2}=\frac{\frac{1}{\sin^{2}x}+\frac{1}{\cos^{2}x}}{(\frac{1}{\sin x}+\frac{1}{\cos x})^2}\]
\[=\frac{\frac{\cos^2 x +\sin^2 x}{\sin^{2}x \cos^2 x}}{(\frac{\cos x+\sin x}{\sin x \cos x})^2}=\frac{\frac{1}{\sin^{2}x \cos^2 x}}{\frac{(\cos x+\sin x)^2}{\sin^2 x \cos^2 x}}\]
\[=\frac{1}{(\sin x+\cos x)^2}=\frac{1}{\sin^2 x+2\sin x \cos x+\cos^2 x}\]
\[=\frac{1}{1+2\sin x \cos x}=(1+2\sin x \cos x)^{-1}.\]