If `sec5A="cosec"(4A-18^(@))and 5A` is an acute angle , find the vlaue of A.

Answer

To solve the equation \( \sec(5A) = \csc(4A - 18^\circ) \) given that \( 5A \) is an acute angle, we can follow these steps:

### Step 1: Write the equation
We start with the equation:
\[ \sec(5A) = \csc(4A - 18^\circ) \]

### Step 2: Use the identity for cosecant
Recall the identity that relates secant and cosecant:
\[ \sec(\theta) = \csc(90^\circ - \theta) \]
Using this identity, we can rewrite the equation:
\[ \sec(5A) = \csc(90^\circ - 5A) \]
Thus, we can equate:
\[ \csc(90^\circ - 5A) = \csc(4A - 18^\circ) \]

### Step 3: Set the angles equal
Since the cosecant function is equal when their angles are equal, we can set up the equation:
\[ 90^\circ - 5A = 4A - 18^\circ \]

### Step 4: Simplify the equation
Now, we will simplify the equation:
\[ 90^\circ + 18^\circ = 4A + 5A \]
This simplifies to:
\[ 108^\circ = 9A \]

### Step 5: Solve for A
Now, we can solve for \( A \):
\[ A = \frac{108^\circ}{9} = 12^\circ \]

### Step 6: Verify the acute angle condition
We need to check if \( 5A \) is an acute angle:
\[ 5A = 5 \times 12^\circ = 60^\circ \]
Since \( 60^\circ \) is indeed an acute angle, our solution is valid.

### Final Answer
Thus, the value of \( A \) is:
\[ \boxed{12^\circ} \]