If ` y = tan ^(-1) (sqrt((1+cos x)/(1-cos x))), " find " (dy)/(dx)`

Answer

To find \(\frac{dy}{dx}\) for the function \(y = \tan^{-1}\left(\sqrt{\frac{1 + \cos x}{1 - \cos x}}\right)\), we can follow these steps:

### Step 1: Rewrite the expression inside the arctangent
We start with the expression inside the inverse tangent function:
\[ y = \tan^{-1}\left(\sqrt{\frac{1 + \cos x}{1 - \cos x}}\right) \]
Using the identities \(1 + \cos x = 2 \cos^2\left(\frac{x}{2}\right)\) and \(1 - \cos x = 2 \sin^2\left(\frac{x}{2}\right)\), we can rewrite the expression:
\[ y = \tan^{-1}\left(\sqrt{\frac{2 \cos^2\left(\frac{x}{2}\right)}{2 \sin^2\left(\frac{x}{2}\right)}}\right) \]
This simplifies to:
\[ y = \tan^{-1}\left(\sqrt{\frac{\cos^2\left(\frac{x}{2}\right)}{\sin^2\left(\frac{x}{2}\right)}}\right) = \tan^{-1}\left(\cot\left(\frac{x}{2}\right)\right) \]

### Step 2: Use the cotangent identity
Using the identity \(\cot \theta = \tan\left(\frac{\pi}{2} - \theta\), we can rewrite \(y\):
\[ y = \tan^{-1}\left(\cot\left(\frac{x}{2}\right)\right) = \frac{\pi}{2} - \frac{x}{2} \]

### Step 3: Differentiate \(y\)
Now, we differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = \frac{d}{dx}\left(\frac{\pi}{2} - \frac{x}{2}\right) \]
Since \(\frac{\pi}{2}\) is a constant, its derivative is \(0\), and the derivative of \(-\frac{x}{2}\) is \(-\frac{1}{2}\):
\[ \frac{dy}{dx} = 0 - \frac{1}{2} = -\frac{1}{2} \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = -\frac{1}{2} \]

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