Differentiate wrt `x` : `(tan x+secx)(cosecx+cot x)`

Options

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

Answer

To differentiate the function \( y = ( \tan x + \sec x )( \csc x + \cot x ) \) with respect to \( x \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u \) and \( v \), then the derivative of their product is given by:

\[ \frac{dy}{dx} = v \frac{du}{dx} + u \frac{dv}{dx} \]

### Step-by-step Solution:

1. **Identify the Functions**:
Let \( u = \tan x + \sec x \) and \( v = \csc x + \cot x \).

2. **Differentiate \( u \)**:
\[ \frac{du}{dx} = \frac{d}{dx}(\tan x) + \frac{d}{dx}(\sec x) \]
Using the derivatives \( \frac{d}{dx}(\tan x) = \sec^2 x \) and \( \frac{d}{dx}(\sec x) = \sec x \tan x \):
\[ \frac{du}{dx} = \sec^2 x + \sec x \tan x \]

3. **Differentiate \( v \)**:
\[ \frac{dv}{dx} = \frac{d}{dx}(\csc x) + \frac{d}{dx}(\cot x) \]
Using the derivatives \( \frac{d}{dx}(\csc x) = -\csc x \cot x \) and \( \frac{d}{dx}(\cot x) = -\csc^2 x \):
\[ \frac{dv}{dx} = -\csc x \cot x - \csc^2 x \]

4. **Apply the Product Rule**:
Now, applying the product rule:
\[ \frac{dy}{dx} = v \frac{du}{dx} + u \frac{dv}{dx} \]
Substituting \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \):
\[ \frac{dy}{dx} = (\csc x + \cot x)(\sec^2 x + \sec x \tan x) + (\tan x + \sec x)(-\csc x \cot x - \csc^2 x) \]

5. **Simplify the Expression**:
Expanding both terms:
\[ \frac{dy}{dx} = (\csc x + \cot x)(\sec^2 x + \sec x \tan x) - (\tan x + \sec x)(\csc x \cot x + \csc^2 x) \]

6. **Factor Common Terms**:
Look for common factors in the expression. We can factor out \( \csc x \cot x \) from the second term and simplify further if possible.

7. **Final Result**:
After simplification, we arrive at the final expression for the derivative:
\[ \frac{dy}{dx} = (\csc x + \cot x)(\sec^2 x + \sec x \tan x) - (\tan x + \sec x)(\csc x \cot x + \csc^2 x) \]