`( cos x )/( 1 + sin x ) + ( 1 + sin x)/( cos x)` is equal to:

Options

A. `2 cos x`
B. `2 sin x`
C. `2 sec x`
D. `2 cosec x`

Answer

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

### Step 1: Combine the fractions
We start by finding a common denominator for the two fractions. The common denominator will be \((1 + \sin x) \cos x\).

\[ \frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x} = \frac{\cos^2 x + (1 + \sin x)^2}{(1 + \sin x) \cos x} \]

### Step 2: Expand the numerator
Now we will expand the numerator \((1 + \sin x)^2\):

\[ (1 + \sin x)^2 = 1 + 2\sin x + \sin^2 x \]

So, the numerator becomes:

\[ \cos^2 x + 1 + 2\sin x + \sin^2 x \]

### Step 3: Use the Pythagorean identity
Using the identity \(\sin^2 x + \cos^2 x = 1\), we can simplify the numerator:

\[ \cos^2 x + \sin^2 x = 1 \]

Thus, the numerator simplifies to:

\[ 1 + 1 + 2\sin x = 2 + 2\sin x \]

### Step 4: Substitute back into the fraction
Now we substitute this back into our fraction:

\[ \frac{2 + 2\sin x}{(1 + \sin x) \cos x} \]

### Step 5: Factor out the common term
We can factor out 2 from the numerator:

\[ \frac{2(1 + \sin x)}{(1 + \sin x) \cos x} \]

### Step 6: Simplify the expression
Now, we can cancel \((1 + \sin x)\) from the numerator and the denominator (assuming \(1 + \sin x \neq 0\)):

\[ \frac{2}{\cos x} \]

### Step 7: Rewrite in terms of secant
Finally, we can rewrite this as:

\[ 2 \sec x \]

### Final Answer
Thus, the expression \(\frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x}\) simplifies to:

\[ \boxed{2 \sec x} \]