If `2cosec^(2)A=x` , then the value of x is

Options

A. `(1)/((1-cosecA))-(1)/((1+cosecA))`
B. `(1)/((1+sinA))+(1)/((1-tanA))`
C. `(1)/((1+cosA))+(1)/((1-tanA))`
D. `(1)/((1-tanA))-(1)/((1+sinA))`

Answer

To solve the equation \( 2 \csc^2 A = x \), we will follow these steps:

### Step 1: Rewrite the equation using trigonometric identities
We know that \( \csc^2 A = \frac{1}{\sin^2 A} \). Therefore, we can rewrite the equation as:
\[ x = 2 \csc^2 A = 2 \cdot \frac{1}{\sin^2 A} = \frac{2}{\sin^2 A} \]

### Step 2: Use the Pythagorean identity
From the Pythagorean identity, we know that \( \sin^2 A + \cos^2 A = 1 \). We can express \( \sin^2 A \) in terms of \( \cos^2 A \):
\[ \sin^2 A = 1 - \cos^2 A \]

### Step 3: Substitute \(\sin^2 A\) back into the equation
Substituting this back into our expression for \( x \):
\[ x = \frac{2}{1 - \cos^2 A} \]

### Step 4: Factor the denominator
We can factor the denominator using the difference of squares:
\[ 1 - \cos^2 A = (1 + \cos A)(1 - \cos A) \]
Thus, we can rewrite \( x \) as:
\[ x = \frac{2}{(1 + \cos A)(1 - \cos A)} \]

### Step 5: Simplify the expression
Now, we can separate the fraction:
\[ x = \frac{2}{1 + \cos A} \cdot \frac{1}{1 - \cos A} \]

### Step 6: Final expression for \( x \)
This gives us the final expression for \( x \):
\[ x = \frac{2}{(1 + \cos A)(1 - \cos A)} \]

### Conclusion
Thus, the value of \( x \) is:
\[ x = \frac{2}{(1 + \cos A)(1 - \cos A)} \]