What is `(sinx)/(1+cosx)+(1+cosx)/(sinx)` equal to ?

Options

A. 2 tan x
B. `2" cosec " x`
C. 2cos x
D. 2sin x

Explanation

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

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

### Step 2: Rewrite the expression
We can rewrite the expression as:
\[ \frac{\sin^2 x}{(1 + \cos x) \sin x} + \frac{(1 + \cos x)^2}{(1 + \cos x) \sin x} \]

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

### Step 4: Expand the numerator
Next, we expand the numerator:
\[ (1 + \cos x)^2 = 1 + 2\cos x + \cos^2 x \]
Thus, the numerator becomes:
\[ \sin^2 x + 1 + 2\cos x + \cos^2 x \]

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

### Step 6: Factor the numerator
Now we can factor the numerator:
\[ 2(1 + \cos x) \]

### Step 7: Substitute back into the expression
Substituting the factored numerator back into the expression gives us:
\[ \frac{2(1 + \cos x)}{(1 + \cos x) \sin x} \]

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

### Step 9: Write the final answer
Thus, the final answer is:
\[ \frac{2}{\sin x} = 2 \csc x \]