If `(1)/((1+sinA))+(1)/(1-(sinA))=x`, then value of x is

Options

A. `2cosec^(2)A`
B. `2sec^(2)A`
C. `2cosecA`
D. `2secA`

Answer

To solve the equation

\[ \frac{1}{1 + \sin A} + \frac{1}{1 - \sin A} = x, \]

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 fractions
We can rewrite the left-hand side using the common denominator:

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

### Step 3: Simplify the numerator
Now, simplify the numerator:

\[ 1 - \sin A + 1 + \sin A = 2. \]

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

\[ \frac{2}{(1 + \sin A)(1 - \sin A)}. \]

### Step 5: Use the identity for the denominator
Recall that \(1 - \sin^2 A = \cos^2 A\). Therefore, we can rewrite the denominator:

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

### Step 6: Substitute back into the expression
Substituting this back into our expression gives:

\[ x = \frac{2}{\cos^2 A}. \]

### Step 7: Use the secant function
We know that \(\frac{1}{\cos^2 A} = \sec^2 A\). Thus, we can write:

\[ x = 2 \sec^2 A. \]

### Final Answer
The value of \(x\) is:

\[ \boxed{2 \sec^2 A}. \]

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