If sec 4A = cosec `(A - 20^(@))`, where 4A is an acute angle, find the value of A.

Answer

To solve the equation \( \sec 4A = \csc (A - 20^\circ) \), where \( 4A \) is an acute angle, we can follow these steps:

### Step 1: Use the complementary angle identity
We know that \( \sec \theta = \csc (90^\circ - \theta) \). Therefore, we can rewrite \( \sec 4A \) as:
\[ \sec 4A = \csc (90^\circ - 4A) \]

### Step 2: Set the two expressions equal
Since we have \( \sec 4A = \csc (A - 20^\circ) \), we can set the two expressions equal:
\[ \csc (90^\circ - 4A) = \csc (A - 20^\circ) \]

### Step 3: Use the property of cosecant
The property of cosecant states that if \( \csc x = \csc y \), then:
\[ x = y + n \cdot 180^\circ \quad \text{or} \quad x = 180^\circ - y + n \cdot 180^\circ \]
for some integer \( n \). Since \( 4A \) is acute, we will consider:
\[ 90^\circ - 4A = A - 20^\circ \]

### Step 4: Solve for A
Now we can solve the equation:
\[ 90^\circ - 4A = A - 20^\circ \]
Rearranging gives:
\[ 90^\circ + 20^\circ = A + 4A \]
\[ 110^\circ = 5A \]
Dividing both sides by 5:
\[ A = \frac{110^\circ}{5} = 22^\circ \]

### Final Answer
Thus, the value of \( A \) is:
\[ \boxed{22^\circ} \]