`cos theta/(1 - sin theta) - costheta/(1 + sin theta) =2`,sin satisfied by which one of the following values of `theta` ?

Options

A. `pi//2`
B. `pi//3`
C. `pi//4`
D. `pi//6`

Answer

To solve the equation

\[ \frac{\cos \theta}{1 - \sin \theta} - \frac{\cos \theta}{1 + \sin \theta} = 2, \]

we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the two fractions on the left side is \((1 - \sin \theta)(1 + \sin \theta)\).

### Step 2: Rewrite the equation
Rewriting the left side using the common denominator, we have:

\[ \frac{\cos \theta(1 + \sin \theta) - \cos \theta(1 - \sin \theta)}{(1 - \sin \theta)(1 + \sin \theta)} = 2. \]

### Step 3: Simplify the numerator
Now, simplify the numerator:

\[ \cos \theta(1 + \sin \theta) - \cos \theta(1 - \sin \theta) = \cos \theta + \cos \theta \sin \theta - \cos \theta + \cos \theta \sin \theta = 2 \cos \theta \sin \theta. \]

So, we can rewrite the equation as:

\[ \frac{2 \cos \theta \sin \theta}{(1 - \sin \theta)(1 + \sin \theta)} = 2. \]

### Step 4: Simplify the equation
Cross-multiplying gives us:

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

### Step 5: Use the Pythagorean identity
Using the identity \(1 - \sin^2 \theta = \cos^2 \theta\), we can rewrite the equation as:

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

### Step 6: Divide by 2
Dividing both sides by 2 (assuming \(\cos \theta \neq 0\)) gives us:

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

### Step 7: Rearranging the equation
Rearranging gives:

\[ \sin \theta = \cos \theta. \]

### Step 8: Solve for \(\theta\)
The equation \(\sin \theta = \cos \theta\) implies:

\[ \tan \theta = 1. \]

### Step 9: Find the angle
The angle \(\theta\) for which \(\tan \theta = 1\) is:

\[ \theta = \frac{\pi}{4} \text{ (or 45 degrees)}. \]

### Conclusion
Thus, the value of \(\theta\) that satisfies the equation is:

\[ \theta = \frac{\pi}{4}. \]