Given: `cos 38^(@)sec(90^(@)-2A)=1,` find the value of angle A.

Answer

To solve the equation \( \cos 38^\circ \sec(90^\circ - 2A) = 1 \) and find the value of angle \( A \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \cos 38^\circ \sec(90^\circ - 2A) = 1 \]

### Step 2: Recall the identity of secant
Recall that \( \sec(x) = \frac{1}{\cos(x)} \). Therefore, we can rewrite the equation as:
\[ \cos 38^\circ \cdot \frac{1}{\cos(90^\circ - 2A)} = 1 \]

### Step 3: Simplify using the identity
Using the co-function identity \( \cos(90^\circ - x) = \sin(x) \), we can rewrite \( \cos(90^\circ - 2A) \) as \( \sin(2A) \):
\[ \cos 38^\circ \cdot \frac{1}{\sin(2A)} = 1 \]

### Step 4: Rearrange the equation
Now, we can rearrange the equation to isolate \( \sin(2A) \):
\[ \cos 38^\circ = \sin(2A) \]

### Step 5: Use the complementary angle identity
Since \( \sin(90^\circ - x) = \cos(x) \), we can express \( \sin(2A) \) in terms of cosine:
\[ \sin(2A) = \sin(90^\circ - 38^\circ) = \sin(52^\circ) \]

### Step 6: Set the angles equal
Now, we have:
\[ 2A = 52^\circ \]

### Step 7: Solve for \( A \)
Dividing both sides by 2 gives us:
\[ A = \frac{52^\circ}{2} = 26^\circ \]

### Final Answer
Thus, the value of angle \( A \) is:
\[ \boxed{26^\circ} \]

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