` (d)/(dx) (cosec x+ cot x )=`

Options

A. ` cosec x(cosec x-cotx ) `
B. ` -cosec x(cosec x-cotx ) `
C. ` cosec x(cosec x+cotx ) `
D. `- cosec x(cosec x+cotx ) `

Answer

To differentiate the expression \( \csc x + \cot x \) with respect to \( x \), we will follow these steps:

### Step 1: Write down the expression to differentiate
We need to differentiate:
\[ y = \csc x + \cot x \]

### Step 2: Differentiate \( \csc x \)
The derivative of \( \csc x \) is:
\[ \frac{d}{dx}(\csc x) = -\csc x \cot x \]

### Step 3: Differentiate \( \cot x \)
The derivative of \( \cot x \) is:
\[ \frac{d}{dx}(\cot x) = -\csc^2 x \]

### Step 4: Combine the derivatives
Now, we combine the derivatives of \( \csc x \) and \( \cot x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(\csc x) + \frac{d}{dx}(\cot x) = -\csc x \cot x - \csc^2 x \]

### Step 5: Factor out common terms
We can factor out \( -\csc x \):
\[ \frac{dy}{dx} = -\csc x (\cot x + \csc x) \]

### Final Answer
Thus, the derivative of \( \csc x + \cot x \) is:
\[ \frac{dy}{dx} = -\csc x (\cot x + \csc x) \]

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