If cosec `(90^@+A)` + x cot A cot `(90^@+A)` = sin `(90^@+A)` , find x.

Answer

To solve the equation \( \csc(90^\circ + A) + x \cot A \cot(90^\circ + A) = \sin(90^\circ + A) \), we will follow these steps:

### Step 1: Simplify the trigonometric functions
We know the following identities:
- \( \csc(90^\circ + A) = \sec A \)
- \( \sin(90^\circ + A) = \cos A \)
- \( \cot(90^\circ + A) = -\tan A \)

Substituting these into the equation gives us:
\[ \sec A + x \cot A (-\tan A) = \cos A \]

### Step 2: Rewrite the equation
Since \( \cot A = \frac{\cos A}{\sin A} \) and \( \tan A = \frac{\sin A}{\cos A} \), we can rewrite \( \cot A \tan A \) as:
\[ \cot A \tan A = \frac{\cos A}{\sin A} \cdot \frac{\sin A}{\cos A} = 1 \]
Thus, the equation simplifies to:
\[ \sec A - x = \cos A \]

### Step 3: Express secant in terms of cosine
Recall that \( \sec A = \frac{1}{\cos A} \). Substituting this in gives:
\[ \frac{1}{\cos A} - x = \cos A \]

### Step 4: Rearranging the equation
Rearranging the equation to isolate \( x \):
\[ \frac{1}{\cos A} - \cos A = x \]

### Step 5: Finding a common denominator
To combine the terms on the left side, we need a common denominator:
\[ x = \frac{1 - \cos^2 A}{\cos A} \]

### Step 6: Using the Pythagorean identity
Using the identity \( 1 - \cos^2 A = \sin^2 A \), we can substitute this into our equation:
\[ x = \frac{\sin^2 A}{\cos A} \]

### Step 7: Final expression for x
Thus, we have:
\[ x = \sin^2 A \cdot \sec A \]
or equivalently,
\[ x = \frac{\sin^2 A}{\cos A} \]

### Conclusion
The value of \( x \) is:
\[ x = \frac{\sin^2 A}{\cos A} \]