If `pi lt theta lt (3pi)/(2)`, then find the value of `sqrt((1-cos2theta)/(1+cos 2 theta))`.

Answer

To solve the problem, we need to find the value of

\[ y = \sqrt{\frac{1 - \cos 2\theta}{1 + \cos 2\theta}} \]

given that \(\pi < \theta < \frac{3\pi}{2}\).

### Step 1: Use the double angle identity for cosine
We know that

\[ \cos 2\theta = 2\cos^2 \theta - 1 \]

and

\[ \cos 2\theta = 1 - 2\sin^2 \theta. \]

We will use both forms in our calculations.

### Step 2: Substitute the identity into the expression
We can rewrite \(y\) using the identity for \(\cos 2\theta\):

\[ y = \sqrt{\frac{1 - (1 - 2\sin^2 \theta)}{1 + (1 - 2\sin^2 \theta)}} \]

This simplifies to:

\[ y = \sqrt{\frac{2\sin^2 \theta}{2 - 2\sin^2 \theta}}. \]

### Step 3: Simplify the expression
Now, we can simplify the fraction:

\[ y = \sqrt{\frac{2\sin^2 \theta}{2(1 - \sin^2 \theta)}} = \sqrt{\frac{\sin^2 \theta}{1 - \sin^2 \theta}}. \]

### Step 4: Use the identity \(1 - \sin^2 \theta = \cos^2 \theta\)
We know that \(1 - \sin^2 \theta = \cos^2 \theta\), so we can substitute this into our expression:

\[ y = \sqrt{\frac{\sin^2 \theta}{\cos^2 \theta}}. \]

### Step 5: Simplify further
This simplifies to:

\[ y = \sqrt{\tan^2 \theta} = |\tan \theta|. \]

### Step 6: Determine the sign of \(\tan \theta\)
Since we are given that \(\pi < \theta < \frac{3\pi}{2}\), we know that in this interval, \(\tan \theta\) is positive. Therefore, we can write:

\[ y = \tan \theta. \]

### Final Answer
Thus, the value of

\[ \sqrt{\frac{1 - \cos 2\theta}{1 + \cos 2\theta}} = \tan \theta. \]

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