If `cosecA+cotA=x` , then value of x is

Options

A. `(1)/((cosec A-cotA))`
B. `(1)/((secA-tanA))`
C. `(1)/((secA-cosA))`
D. `(1)/((sinA-cosA))`

Answer

To solve the equation \( \csc A + \cot A = x \), we can follow these steps:

### Step 1: Rewrite the Trigonometric Functions
We know that:
\[ \csc A = \frac{1}{\sin A} \quad \text{and} \quad \cot A = \frac{\cos A}{\sin A} \]
So, we can rewrite the equation as:
\[ \csc A + \cot A = \frac{1}{\sin A} + \frac{\cos A}{\sin A} \]

### Step 2: Combine the Terms
Since both terms have a common denominator of \( \sin A \), we can combine them:
\[ \csc A + \cot A = \frac{1 + \cos A}{\sin A} \]
Thus, we have:
\[ x = \frac{1 + \cos A}{\sin A} \]

### Step 3: Rationalize the Expression
To simplify further, we can multiply the numerator and the denominator by \( 1 - \cos A \):
\[ x = \frac{(1 + \cos A)(1 - \cos A)}{\sin A(1 - \cos A)} \]

### Step 4: Apply the Difference of Squares
Using the difference of squares in the numerator:
\[ 1 + \cos A)(1 - \cos A) = 1 - \cos^2 A \]
Thus, we can rewrite \( x \) as:
\[ x = \frac{1 - \cos^2 A}{\sin A(1 - \cos A)} \]

### Step 5: Use the Pythagorean Identity
We know from the Pythagorean identity that:
\[ 1 - \cos^2 A = \sin^2 A \]
Substituting this into our equation gives:
\[ x = \frac{\sin^2 A}{\sin A(1 - \cos A)} \]

### Step 6: Simplify the Expression
Now, we can simplify:
\[ x = \frac{\sin A}{1 - \cos A} \]

### Step 7: Final Expression
Thus, the value of \( x \) is:
\[ x = \frac{\sin A}{1 - \cos A} \]