If \(\mathrm{f}(x)=\frac{\mathrm{a} \sin x+b \cos x}{\mathrm{c} \sin x+\mathrm{d} \cos x}\) is decreasing for all \(x\), then

\(\begin{aligned} & \mathrm{f}(x)=\frac{\mathrm{a} \sin x+\mathrm{b} \cos x}{\mathrm{c} \sin x+\mathrm{d} \cos x} \\ & \mathrm{f}(x) \text { will be decreasing, if } \mathrm{f}^{\prime}(x) \lt 0 \\ \therefore \quad & \frac{1}{(\operatorname{csin} x+\mathrm{d} \cos x)^2}[(\mathrm{csin} x+\mathrm{d} \cos x)(\mathrm{a} \cos x-b \sin x)\end{aligned}\)
\(-(a \sin x+b \cos x)(\cos x-d \sin x)] \lt 0\)
\(\begin{aligned} & \Rightarrow \operatorname{acsin} x \cos x-\mathrm{bc} \sin ^2 x+\mathrm{ad} \cos ^2 x \\ & \quad-\mathrm{bd} \sin x \cos x-\mathrm{acsin} x \cos x+\mathrm{ad} \sin ^2 x\end{aligned}\)
\(-\mathrm{bc} \cos ^2 x+\mathrm{bdsin} x \cos x \lt 0\)
\(\begin{aligned} & \Rightarrow \mathrm{ad}\left(\sin ^2 x+\cos ^2 x\right)-\mathrm{bc}\left(\sin ^2 x+\cos ^2 x\right) \lt 0 \\ & \Rightarrow \mathrm{ad}-\mathrm{bc} \lt 0\end{aligned}\)