The value of `(sinA)/(1+cosA)+(sinA)/(1-cosA)` is `(0^(@)ltAlt90^(@))`

Options

A. `2 "cosec"A`
B. `2 secA`
C. `2 sin A`
D. `2 cosA`

Answer

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

### Step 1: Combine the fractions
We start by finding a common denominator for the two fractions. The common denominator is \((1 + \cos A)(1 - \cos A)\).

\[ \frac{\sin A}{1 + \cos A} + \frac{\sin A}{1 - \cos A} = \frac{\sin A(1 - \cos A) + \sin A(1 + \cos A)}{(1 + \cos A)(1 - \cos A)} \]

### Step 2: Simplify the numerator
Now, we simplify the numerator:

\[ \sin A(1 - \cos A) + \sin A(1 + \cos A) = \sin A - \sin A \cos A + \sin A + \sin A \cos A \]

Combining like terms:

\[ = \sin A + \sin A = 2 \sin A \]

### Step 3: Simplify the denominator
Next, we simplify the denominator using the difference of squares:

\[ (1 + \cos A)(1 - \cos A) = 1 - \cos^2 A \]

Using the Pythagorean identity, we know that \(1 - \cos^2 A = \sin^2 A\).

### Step 4: Write the complete expression
Now we can write the complete expression as:

\[ \frac{2 \sin A}{\sin^2 A} \]

### Step 5: Simplify the fraction
We can simplify this fraction:

\[ \frac{2 \sin A}{\sin^2 A} = \frac{2}{\sin A} \]

### Step 6: Final result
Thus, the final result is:

\[ 2 \csc A \]