Among given values of `theta` which one satisfies the equation `(cos theta)/(1- sin theta) -(cos theta)/(1 + sin theta) =2` ?

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: Combine the fractions on the left-hand side
We can combine the two fractions by finding a common denominator. The common denominator is \((1 - \sin \theta)(1 + \sin \theta)\). Thus, we rewrite the left-hand side:

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

### Step 2: Simplify the numerator
Now, simplify the numerator:

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

### Step 3: Substitute back into the equation
Now, substituting back into the equation gives us:

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

### Step 4: Cross-multiply
Cross-multiplying gives:

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

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

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

### Step 6: Divide by 2
Dividing both sides by 2 simplifies to:

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

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

\[ \cos \theta \sin \theta - \cos^2 \theta = 0. \]

### Step 8: Factor out \(\cos \theta\)
Factoring out \(\cos \theta\) gives:

\[ \cos \theta (\sin \theta - \cos \theta) = 0. \]

### Step 9: Set each factor to zero
This gives us two cases:

1. \(\cos \theta = 0\)
2. \(\sin \theta - \cos \theta = 0\) or \(\sin \theta = \cos \theta\).

### Step 10: Solve for \(\theta\)
1. From \(\cos \theta = 0\), we have \(\theta = \frac{\pi}{2} + n\pi\) for \(n \in \mathbb{Z}\).
2. From \(\sin \theta = \cos \theta\), we have \(\tan \theta = 1\), which gives \(\theta = \frac{\pi}{4} + n\pi\) for \(n \in \mathbb{Z}\).

### Conclusion
Among the given values of \(\theta\), the one that satisfies the equation is \(\theta = \frac{\pi}{4}\) (which corresponds to 45 degrees).

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