The value of `(sintheta)/(1+costheta)+(sintheta)/(1-costheta)` is :

Options

A. `2sintheta`
B. `2costheta`
C. `2sectheta`
D. `2cosectheta`

Answer

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

### Step 1: Write the expression
We start with the expression:
\[ \frac{\sin \theta}{1 + \cos \theta} + \frac{\sin \theta}{1 - \cos \theta} \]

### Step 2: Find a common denominator
The common denominator for the two fractions is \((1 + \cos \theta)(1 - \cos \theta)\). Thus, we can rewrite the expression as:
\[ \frac{\sin \theta (1 - \cos \theta) + \sin \theta (1 + \cos \theta)}{(1 + \cos \theta)(1 - \cos \theta)} \]

### Step 3: Simplify the numerator
Now, simplify the numerator:
\[ \sin \theta (1 - \cos \theta) + \sin \theta (1 + \cos \theta) = \sin \theta - \sin \theta \cos \theta + \sin \theta + \sin \theta \cos \theta \]
Combining like terms:
\[ = \sin \theta + \sin \theta = 2\sin \theta \]

### Step 4: Simplify the denominator
The denominator can be simplified using the difference of squares:
\[ (1 + \cos \theta)(1 - \cos \theta) = 1 - \cos^2 \theta = \sin^2 \theta \]

### Step 5: Combine the results
Now we can combine the simplified numerator and denominator:
\[ \frac{2 \sin \theta}{\sin^2 \theta} \]

### Step 6: Simplify the fraction
This simplifies to:
\[ \frac{2}{\sin \theta} = 2 \csc \theta \]

### Step 7: Use the identity
Since \(\csc \theta = \frac{1}{\sin \theta}\), we can also express this in terms of cosine:
\[ = 2 \cdot \frac{1}{\sin \theta} = 2 \cdot \frac{\cos \theta}{\sin \theta \cdot \cos \theta} = 2 \cos \theta \]

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