`Ify=( sec x^(@) +tan x^(@))/( sec ^(@) -tan x^(@) ) ,then (dy)/(dx) `

Options

A. `(pi)/(180) tan ((pi)/( 4)+(x^(@) )/( 2))sec ^(2) ((pi)/(4) + (x^(@))/( 2))`
B. `(-pi)/(180) tan ((pi)/( 4)+(x^(@) )/( 2))sec ^(2) ((pi)/(4) + (x^(@))/( 2))`
C. `(-pi)/(360) tan ((pi)/( 4)+(x^(@) )/( 2))sec ^(2) ((pi)/(4) + (x^(@))/( 2))`
D. `(-pi)/(90) tan ((pi)/( 4)+(x^(@) )/( 2))sec ^(2) ((pi)/(4) + (x^(@))/( 2))`

Answer

To find the derivative \( \frac{dy}{dx} \) for the function

\[ y = \frac{\sec^2 x + \tan^2 x}{\sec^2 x - \tan^2 x} \]

we can follow these steps:

### Step 1: Simplify the Function
First, we multiply and divide the numerator and denominator by \( \sec x + \tan x \):

\[ y = \frac{(\sec^2 x + \tan^2 x)(\sec x + \tan x)}{(\sec^2 x - \tan^2 x)(\sec x + \tan x)} \]

### Step 2: Simplify the Denominator
Using the identity \( \sec^2 x - \tan^2 x = 1 \), we can simplify the denominator:

\[ y = \frac{(\sec^2 x + \tan^2 x)(\sec x + \tan x)}{1 \cdot (\sec x + \tan x)} = \sec^2 x + \tan^2 x \]

### Step 3: Differentiate the Function
Now we differentiate \( y \):

\[ \frac{dy}{dx} = \frac{d}{dx}(\sec^2 x + \tan^2 x) \]

Using the derivatives \( \frac{d}{dx}(\sec^2 x) = 2 \sec^2 x \tan x \) and \( \frac{d}{dx}(\tan^2 x) = 2 \tan x \sec^2 x \):

\[ \frac{dy}{dx} = 2 \sec^2 x \tan x + 2 \tan x \sec^2 x = 2 \tan x \sec^2 x + 2 \tan x \sec^2 x = 4 \tan x \sec^2 x \]

### Step 4: Final Result
Thus, the derivative is:

\[ \frac{dy}{dx} = 4 \tan x \sec^2 x \]