`(cos theta)/(1 - sin theta) + (cos theta)/( 1 + sin theta)` equals

Options

A. `2 tan theta `
B. ` 2 cosec theta`
C. ` 2 cot theta`
D. `2 sec theta `

Answer

To solve the expression \(\frac{\cos \theta}{1 - \sin \theta} + \frac{\cos \theta}{1 + \sin \theta}\), we will follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \frac{\cos \theta}{1 - \sin \theta} + \frac{\cos \theta}{1 + \sin \theta} \]

### Step 2: Find a common denominator
The common denominator for the two fractions is \((1 - \sin \theta)(1 + \sin \theta)\). Thus, we can rewrite the expression as:
\[ \frac{\cos \theta(1 + \sin \theta)}{(1 - \sin \theta)(1 + \sin \theta)} + \frac{\cos \theta(1 - \sin \theta)}{(1 - \sin \theta)(1 + \sin \theta)} \]

### Step 3: Combine the fractions
Now, we can combine the two fractions:
\[ \frac{\cos \theta(1 + \sin \theta) + \cos \theta(1 - \sin \theta)}{(1 - \sin \theta)(1 + \sin \theta)} \]

### Step 4: Simplify the numerator
In the numerator, we can factor out \(\cos \theta\):
\[ \cos \theta((1 + \sin \theta) + (1 - \sin \theta)) = \cos \theta(1 + \sin \theta + 1 - \sin \theta) = \cos \theta(2) \]
So, the numerator simplifies to:
\[ 2 \cos \theta \]

### Step 5: Simplify the denominator
The denominator can be simplified using the difference of squares:
\[ (1 - \sin \theta)(1 + \sin \theta) = 1^2 - \sin^2 \theta = 1 - \sin^2 \theta \]
Using the Pythagorean identity, we know that \(1 - \sin^2 \theta = \cos^2 \theta\).

### Step 6: Write the final expression
Now we can write the entire expression as:
\[ \frac{2 \cos \theta}{\cos^2 \theta} \]

### Step 7: Simplify further
We can simplify this by canceling one \(\cos \theta\) from the numerator and the denominator:
\[ \frac{2}{\cos \theta} \]

### Step 8: Final result
This can be rewritten using the secant function:
\[ 2 \sec \theta \]

Thus, the final answer is:
\[ 2 \sec \theta \]

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