The function frac{asin x + bcos x}{csin x + d | vedclass

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(A) Let $y = \frac{a\sin x + b\cos x}{c\sin x + d\cos x}$.The function is decreasing when $\frac{dy}{dx} < 0$.Using the quotient rule,$\frac{dy}{dx} =

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$ad - bc < 0$

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(A) Let $y = \frac{a\sin x + b\cos x}{c\sin x + d\cos x}$.The function is decreasing when $\frac{dy}{dx} Using the quotient rule,$\frac{dy}{dx} = \frac{(c\sin x + d\cos x)(a\cos x - b\sin x) - (a\sin x + b\cos x)(c\cos x - d\sin x)}{(c\sin x + d\cos x)^2}$.Expanding the numerator:$= (ac\sin x \cos x - bc\sin^2 x + ad\cos^2 x - bd\sin x \cos x) - (ac\sin x \cos x - ad\sin^2 x + bc\cos^2 x - bd\sin x \cos x)$.Simplifying the expression:$= ad(\cos^2 x + \sin^2 x) - bc(\sin^2 x + \cos^2 x) = ad - bc$.Since the denominator $(c\sin x + d\cos x)^2$ is always positive,the condition $\frac{dy}{dx} < 0$ implies $ad - bc < 0$.

The function $\frac{a\sin x + b\cos x}{c\sin x + d\cos x}$ is decreasing,if

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