Find the value of the x which will satisfy sec A cosec `(90^@-A)-x cot (90^@-A)=1`

Answer

To solve the equation \( \sec A \csc (90^\circ - A) - x \cot (90^\circ - A) = 1 \), we will follow these steps:

### Step 1: Use Trigonometric Identities
We know the following trigonometric identities:
- \( \csc(90^\circ - A) = \sec A \)
- \( \cot(90^\circ - A) = \tan A \)

Substituting these identities into the equation gives us:
\[ \sec A \cdot \sec A - x \cdot \tan A = 1 \]

### Step 2: Simplify the Equation
This simplifies to:
\[ \sec^2 A - x \tan A = 1 \]

### Step 3: Rearrange the Equation
Rearranging the equation to isolate \( x \tan A \):
\[ \sec^2 A - 1 = x \tan A \]

### Step 4: Use the Pythagorean Identity
We know from the Pythagorean identity that:
\[ \sec^2 A - 1 = \tan^2 A \]
Substituting this into our equation gives:
\[ \tan^2 A = x \tan A \]

### Step 5: Factor the Equation
We can factor the equation:
\[ \tan A (\tan A - x) = 0 \]

### Step 6: Solve for \( x \)
This gives us two possible solutions:
1. \( \tan A = 0 \) (which is not a solution for \( x \))
2. \( \tan A - x = 0 \) or \( x = \tan A \)

Thus, the value of \( x \) that satisfies the original equation is:
\[ x = \tan A \]

### Final Answer
The value of \( x \) is \( \tan A \).

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