If ` y = sqrt((1 + cos 2x)/(1-cos 2x))" find " (dy)/(dx) `

Answer

To find \(\frac{dy}{dx}\) for the function \(y = \sqrt{\frac{1 + \cos 2x}{1 - \cos 2x}}\), we can follow these steps:

### Step 1: Simplify the expression
We start with the given expression:
\[ y = \sqrt{\frac{1 + \cos 2x}{1 - \cos 2x}} \]

Using trigonometric identities, we know:
\[ 1 + \cos 2x = 2 \cos^2 x \quad \text{and} \quad 1 - \cos 2x = 2 \sin^2 x \]

Substituting these identities into the equation gives:
\[ y = \sqrt{\frac{2 \cos^2 x}{2 \sin^2 x}} = \sqrt{\frac{\cos^2 x}{\sin^2 x}} = \sqrt{\cot^2 x} \]

Since the square root of \(\cot^2 x\) is \(|\cot x|\), we can simplify further:
\[ y = |\cot x| \]

### Step 2: Differentiate \(y\)
To find \(\frac{dy}{dx}\), we differentiate \(y = \cot x\) (assuming \(x\) is in the range where \(\cot x\) is positive):
\[ \frac{dy}{dx} = \frac{d}{dx}(\cot x) \]

Using the derivative of \(\cot x\):
\[ \frac{dy}{dx} = -\csc^2 x \]

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

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