`(1 + sin theta)/(cos theta) + (cos theta)/(1 + sin theta)` is equal to

Options

A. ` sec theta`
B. `2 sec theta`
C. `2 cos theta`
D. `sin theta`

Answer

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

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

### Step 2: Rewrite each fraction with the common denominator
We rewrite the fractions:
\[ \frac{1 + \sin \theta}{\cos \theta} = \frac{(1 + \sin \theta)(1 + \sin \theta)}{\cos \theta(1 + \sin \theta)} = \frac{(1 + \sin \theta)^2}{\cos \theta(1 + \sin \theta)} \]
\[ \frac{\cos \theta}{1 + \sin \theta} = \frac{\cos \theta \cos \theta}{\cos \theta(1 + \sin \theta)} = \frac{\cos^2 \theta}{\cos \theta(1 + \sin \theta)} \]

### Step 3: Combine the fractions
Now we can combine the two fractions:
\[ \frac{(1 + \sin \theta)^2 + \cos^2 \theta}{\cos \theta(1 + \sin \theta)} \]

### Step 4: Expand the numerator
Expanding \((1 + \sin \theta)^2\):
\[ (1 + \sin \theta)^2 = 1 + 2\sin \theta + \sin^2 \theta \]
So the numerator becomes:
\[ 1 + 2\sin \theta + \sin^2 \theta + \cos^2 \theta \]

### Step 5: Use the Pythagorean identity
Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), we can simplify:
\[ 1 + 2\sin \theta + 1 = 2 + 2\sin \theta \]

### Step 6: Substitute back into the fraction
Now substituting back into our fraction:
\[ \frac{2 + 2\sin \theta}{\cos \theta(1 + \sin \theta)} \]

### Step 7: Factor out the common term in the numerator
Factoring out 2 from the numerator:
\[ \frac{2(1 + \sin \theta)}{\cos \theta(1 + \sin \theta)} \]

### Step 8: Cancel the common terms
We can cancel \(1 + \sin \theta\) from the numerator and denominator (assuming \(1 + \sin \theta \neq 0\)):
\[ \frac{2}{\cos \theta} \]

### Step 9: Use the secant identity
We know that \(\frac{1}{\cos \theta} = \sec \theta\), so:
\[ \frac{2}{\cos \theta} = 2 \sec \theta \]

### Final Answer:
Thus, the expression simplifies to:
\[ 2 \sec \theta \]

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