What is `(1+sinA)/(1-sinA)-(1-sinA)/(1+sinA)`equal to ?

Options

A. `secA-tanA`
B. `2secA.tanA`
C. `4secA.tanA`
D. `4cosecA.cotA`

Explanation

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

### Step 1: Write the expression
We start with the expression:
\[ x = \frac{1 + \sin A}{1 - \sin A} - \frac{1 - \sin A}{1 + \sin A} \]

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

### Step 3: Expand the numerators
Now we expand the squares in the numerator:
\[ (1 + \sin A)^2 = 1 + 2\sin A + \sin^2 A \]
\[ (1 - \sin A)^2 = 1 - 2\sin A + \sin^2 A \]
Substituting these back into the expression gives:
\[ x = \frac{(1 + 2\sin A + \sin^2 A) - (1 - 2\sin A + \sin^2 A)}{(1 - \sin A)(1 + \sin A)} \]

### Step 4: Simplify the numerator
Now simplify the numerator:
\[ x = \frac{1 + 2\sin A + \sin^2 A - 1 + 2\sin A - \sin^2 A}{(1 - \sin A)(1 + \sin A)} \]
This simplifies to:
\[ x = \frac{4\sin A}{(1 - \sin A)(1 + \sin A)} \]

### Step 5: Simplify the denominator
The denominator can be simplified using the identity \(1 - \sin^2 A = \cos^2 A\):
\[ (1 - \sin A)(1 + \sin A) = 1 - \sin^2 A = \cos^2 A \]
Thus, we have:
\[ x = \frac{4\sin A}{\cos^2 A} \]

### Step 6: Convert to sec and tan
Now we can express this in terms of \(\sec A\) and \(\tan A\):
\[ x = 4 \cdot \frac{\sin A}{\cos^2 A} = 4 \tan A \sec A \]

### Final Answer
Therefore, the expression \(\frac{1 + \sin A}{1 - \sin A} - \frac{1 - \sin A}{1 + \sin A}\) is equal to:
\[ \boxed{4 \sec A \tan A} \]