What is the value of `(1+sinA)/(1-sinA)-(1-sinA)/(1+sinA)`?

Options

A. sec A - tan A
B. 2sec A tan A
C. 4sec A. tan A
D. 4cosec A.cot A

Answer

To solve the expression \((1 + \sin A)/(1 - \sin A) - (1 - \sin A)/(1 + \sin A)\), we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the two fractions is \((1 - \sin A)(1 + \sin A)\).

### Step 2: Rewrite the expression
We can rewrite the expression as:
\[ \frac{(1 + \sin A)^2 - (1 - \sin A)^2}{(1 - \sin A)(1 + \sin A)} \]

### Step 3: Expand the numerators
Now, we will expand the numerators:
\[ (1 + \sin A)^2 = 1 + 2\sin A + \sin^2 A \]
\[ (1 - \sin A)^2 = 1 - 2\sin A + \sin^2 A \]

### Step 4: Subtract the expanded forms
Now we subtract the two expanded forms:
\[ (1 + 2\sin A + \sin^2 A) - (1 - 2\sin A + \sin^2 A) \]
This simplifies to:
\[ 1 + 2\sin A + \sin^2 A - 1 + 2\sin A - \sin^2 A = 4\sin A \]

### Step 5: Substitute back into the fraction
Now we substitute back into the fraction:
\[ \frac{4\sin A}{(1 - \sin A)(1 + \sin A)} \]

### Step 6: Simplify the denominator
The denominator can be simplified using the difference of squares:
\[ (1 - \sin A)(1 + \sin A) = 1 - \sin^2 A = \cos^2 A \]

### Step 7: Final expression
Now we have:
\[ \frac{4\sin A}{\cos^2 A} \]
This can be rewritten as:
\[ 4\sec^2 A \cdot \sin A \]

### Step 8: Express in terms of tan
Since \(\tan A = \frac{\sin A}{\cos A}\), we can express this as:
\[ 4\sec A \cdot \tan A \]

### Conclusion
Thus, the final value of the expression is:
\[ 4\sec A \cdot \tan A \]