If `y=tan^(-1)[(sinx+cosx)/(cosx-sinx)]` then `(dy)/(dx)` is

Options

A. `1/2`
B. `(pi)/4`
C. 0
D. 1

Explanation

To solve the problem \( y = \tan^{-1}\left(\frac{\sin x + \cos x}{\cos x - \sin x}\right) \) and find \( \frac{dy}{dx} \), we will follow these steps:

### Step 1: Rewrite the Expression
We start with the given expression:
\[ y = \tan^{-1}\left(\frac{\sin x + \cos x}{\cos x - \sin x}\right) \]

### Step 2: Simplify the Argument of the Inverse Tangent
To simplify the argument, we divide both the numerator and the denominator by \( \cos x \):
\[ y = \tan^{-1}\left(\frac{\frac{\sin x}{\cos x} + 1}{1 - \frac{\sin x}{\cos x}}\right) \]
This simplifies to:
\[ y = \tan^{-1\left(\frac{\tan x + 1}{1 - \tan x}\right) \]

### Step 3: Recognize the Tangent Addition Formula
Recall the tangent addition formula:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
Here, we can identify \( a = x \) and \( b = \frac{\pi}{4} \) (since \( \tan\left(\frac{\pi}{4}\right) = 1 \)):
\[ y = \tan^{-1}\left(\tan\left(x + \frac{\pi}{4}\right)\right) \]

### Step 4: Simplify \( y \)
Since \( \tan^{-1}(\tan(\theta)) = \theta \) for \( \theta \) in the principal range of \( \tan^{-1} \):
\[ y = x + \frac{\pi}{4} \]

### Step 5: Differentiate \( y \)
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}\left(x + \frac{\pi}{4}\right) = 1 \]

### Final Result
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 1 \]

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