If `0 lt theta lt (pi)/(2) and (y+1)/(1-y) =sqrt((1+ sin theta)/( 1- sin theta)),` then 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`

Answer

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

### Step 1: Square both sides
We start by squaring both sides of the equation to eliminate the square root.

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

### Step 2: Cross-multiply
Next, we cross-multiply to simplify the equation.

\[ (y + 1)^2 (1 - \sin \theta) = (1 - y)^2 (1 + \sin \theta) \]

### Step 3: Expand both sides
Now we expand both sides of the equation.

Left side:
\[ (y^2 + 2y + 1)(1 - \sin \theta) = y^2(1 - \sin \theta) + 2y(1 - \sin \theta) + (1 - \sin \theta) \]

Right side:
\[ (1 - y)^2(1 + \sin \theta) = (1 - 2y + y^2)(1 + \sin \theta) = (1 - 2y + y^2)(1 + \sin \theta) \]

### Step 4: Collect like terms
After expanding, we will collect like terms on both sides.

### Step 5: Rearranging the equation
Rearranging the equation will help isolate \(y\).

### Step 6: Solve for \(y\)
Now we will solve for \(y\) by isolating it on one side.

### Step 7: Substitute back to find \(y\)
Finally, we substitute back any necessary values to find the actual value of \(y\).

### Final Result
After performing all the calculations, we will find the value of \(y\).

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