Question

If \[ y=\frac{a\cos x+b\sin x+c}{\sin x}, \] then \[ \frac{dy}{dx}= \]

• A. \(-a\cosec^2x-c\cosec x\cot x\)
• B. \(-a\)
• C. \(-a\cosec^2x+b\sec^2x+c\cosec x\cot x\)
• D. \(a\cosec^2x-c\cosec x\cot x\)

Hint:

Before differentiating, simplify trigonometric fractions into \(\cot x\), \(\tan x\), \(\sec x\), or \(\cosec x\).

Answer 1

The Correct Option is A

Concept: First simplify the given function using trigonometric identities: \[ \frac{\cos x}{\sin x}=\cot x,\qquad \frac{1}{\sin x}=\cosec x \]

Step 1: Given: \[ y=\frac{a\cos x+b\sin x+c}{\sin x} \] Split the fraction: \[ y=\frac{a\cos x}{\sin x}+\frac{b\sin x}{\sin x}+\frac{c}{\sin x} \] \[ y=a\cot x+b+c\cosec x \]

Step 2: Differentiate both sides with respect to \(x\). \[ \frac{dy}{dx} = a\frac{d}{dx}(\cot x)+\frac{d}{dx}(b)+c\frac{d}{dx}(\cosec x) \]

Step 3: Use derivative formulas. \[ \frac{d}{dx}(\cot x)=-\cosec^2x \] and \[ \frac{d}{dx}(\cosec x)=-\cosec x\cot x \]

Step 4: Substitute these values. \[ \frac{dy}{dx} = a(-\cosec^2x)+0+c(-\cosec x\cot x) \] \[ \frac{dy}{dx} = -a\cosec^2x-c\cosec x\cot x \] Therefore, \[ \boxed{-a\cosec^2x-c\cosec x\cot x} \]