Solve : `(sin theta)/(1+cos theta)+(1+cos theta)/(sin theta)=?`
A. `tan theta`
B. `cot theta`
C. `(2)/(sin theta)`
D. `(2)/(cos theta)`

Options

A. A
B. D
C. B
D. C

Answer

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

### Step 1: Find a common denominator
The common denominator for the two fractions is \((1 + \cos \theta) \sin \theta\). We can rewrite the expression as follows:

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

### Step 2: Expand the numerator
Now, we expand the numerator:

\[ (1 + \cos \theta)^2 = 1 + 2\cos \theta + \cos^2 \theta \]

So, the numerator becomes:

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

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

\[ 1 + 1 + 2\cos \theta = 2 + 2\cos \theta \]

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

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

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

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

### Step 6: Cancel the common terms
We can cancel \((1 + \cos \theta)\) from the numerator and denominator (assuming \(1 + \cos \theta \neq 0\)):

\[ \frac{2}{\sin \theta} \]

### Final Result
Thus, the simplified expression is:

\[ \frac{2}{\sin \theta} \]

This corresponds to option C.