If `0lt thetalt(pi)/(2')and if(y+1)/(1-y)=sqrt((1+sintheta)/(1-sintheta'))` they y is equal to

Options

A. `cot""(theta)/(2)`
B. `tan""(theta)/(2)`
C. `cot""(theta)/(2)+tan""(theta)/(2)`
D. `cot""(theta)/(2)-tan""(theta)/(2)`

Explanation

To solve the equation \(\frac{y + 1}{1 - y} = \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}}\), we will follow these steps:

### Step 1: Rewrite the Right Side
We start with the equation:
\[ \frac{y + 1}{1 - y} = \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} \]

### Step 2: Simplify the Right Side
Using the identity \(1 = \cos^2 \frac{\theta}{2} + \sin^2 \frac{\theta}{2}\), we can rewrite \(1 + \sin \theta\) and \(1 - \sin \theta\):
\[ 1 + \sin \theta = \cos^2 \frac{\theta}{2} + \sin^2 \frac{\theta}{2} + 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} = \left(\cos \frac{\theta}{2} + \sin \frac{\theta}{2}\right)^2 \]
\[ 1 - \sin \theta = \cos^2 \frac{\theta}{2} + \sin^2 \frac{\theta}{2} - 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} = \left(\cos \frac{\theta}{2} - \sin \frac{\theta}{2}\right)^2 \]

Thus, we can rewrite the right side:
\[ \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} = \frac{\cos \frac{\theta}{2} + \sin \frac{\theta}{2}}{\cos \frac{\theta}{2} - \sin \frac{\theta}{2}} \]

### Step 3: Set the Left Side Equal to the Right Side
Now we have:
\[ \frac{y + 1}{1 - y} = \frac{\cos \frac{\theta}{2} + \sin \frac{\theta}{2}}{\cos \frac{\theta}{2} - \sin \frac{\theta}{2}} \]

### Step 4: Cross-Multiply
Cross-multiplying gives us:
\[ (y + 1)(\cos \frac{\theta}{2} - \sin \frac{\theta}{2}) = (1 - y)(\cos \frac{\theta}{2} + \sin \frac{\theta}{2}) \]

### Step 5: Expand Both Sides
Expanding both sides:
\[ y \cos \frac{\theta}{2} - y \sin \frac{\theta}{2} + \cos \frac{\theta}{2} - \sin \frac{\theta}{2} = \cos \frac{\theta}{2} - y \cos \frac{\theta}{2} + y \sin \frac{\theta}{2} - \sin \frac{\theta}{2} \]

### Step 6: Combine Like Terms
Rearranging gives:
\[ y \cos \frac{\theta}{2} + y \sin \frac{\theta}{2} = \cos \frac{\theta}{2} - \sin \frac{\theta}{2} + y \sin \frac{\theta}{2} - \sin \frac{\theta}{2} \]

### Step 7: Isolate y
Combining terms leads us to:
\[ y (\cos \frac{\theta}{2} + \sin \frac{\theta}{2}) = \cos \frac{\theta}{2} - \sin \frac{\theta}{2} \]
Thus:
\[ y = \frac{\cos \frac{\theta}{2} - \sin \frac{\theta}{2}}{\cos \frac{\theta}{2} + \sin \frac{\theta}{2}} \]

### Step 8: Recognize the Tangent Half-Angle Identity
The expression simplifies to:
\[ y = \tan \frac{\theta}{2} \]

### Final Answer
Thus, the value of \(y\) is:
\[ \boxed{\tan \frac{\theta}{2}} \]

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