What is the value of `(sin theta)/(1 +cos theta) + (1 + cos theta)/sin theta` ?

Options

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

Answer

To find the value of the expression \((\sin \theta)/(1 + \cos \theta) + (1 + \cos \theta)/\sin \theta\), we can follow these steps:

### Step 1: Combine the fractions
We start by finding a common denominator for the two fractions:
\[ \frac{\sin \theta}{1 + \cos \theta} + \frac{1 + \cos \theta}{\sin \theta} = \frac{\sin^2 \theta + (1 + \cos \theta)^2}{\sin \theta (1 + \cos \theta)} \]

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

### Step 3: Use the Pythagorean identity
Using the 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:
\[ \frac{2 + 2\cos \theta}{\sin \theta (1 + \cos \theta)} \]

### Step 5: Factor out the common terms
We can factor out a 2 from the numerator:
\[ \frac{2(1 + \cos \theta)}{\sin \theta (1 + \cos \theta)} \]

### Step 6: Cancel the common terms
Now we can cancel \(1 + \cos \theta\) from the numerator and the denominator (assuming \(1 + \cos \theta \neq 0\)):
\[ \frac{2}{\sin \theta} \]

### Step 7: Rewrite in terms of cosecant
Finally, we can express this as:
\[ 2 \cdot \csc \theta \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{2 \csc \theta} \]

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