If cos 2A = sin `(A - 15^(@))`, find A.

Answer

To solve the equation \( \cos 2A = \sin (A - 15^\circ) \), we can follow these steps:

### Step 1: Rewrite \( \cos 2A \)
We know that \( \cos 2A \) can be expressed in terms of sine:
\[ \cos 2A = \sin(90^\circ - 2A) \]
So we can rewrite the equation as:
\[ \sin(90^\circ - 2A) = \sin(A - 15^\circ) \]

### Step 2: Set up the sine equation
Since \( \sin x = \sin y \) implies that:
1. \( x = y + n \cdot 360^\circ \) or
2. \( x = 180^\circ - y + n \cdot 360^\circ \) for some integer \( n \)

We can apply this to our equation:
\[ 90^\circ - 2A = A - 15^\circ + n \cdot 360^\circ \quad \text{(1)} \]
or
\[ 90^\circ - 2A = 180^\circ - (A - 15^\circ) + n \cdot 360^\circ \quad \text{(2)} \]

### Step 3: Solve equation (1)
From equation (1):
\[ 90^\circ - 2A = A - 15^\circ \]
Rearranging gives:
\[ 90^\circ + 15^\circ = A + 2A \]
\[ 105^\circ = 3A \]
Dividing both sides by 3:
\[ A = 35^\circ \]

### Step 4: Solve equation (2)
From equation (2):
\[ 90^\circ - 2A = 180^\circ - A + 15^\circ \]
Rearranging gives:
\[ 90^\circ - 2A = 195^\circ - A \]
\[ A - 2A = 195^\circ - 90^\circ \]
\[ -A = 105^\circ \]
Dividing by -1:
\[ A = -105^\circ \]
Since angles are typically considered in the range of \(0^\circ\) to \(360^\circ\), we discard \(A = -105^\circ\) as it is not a valid solution in this context.

### Final Answer
Thus, the only valid solution is:
\[ A = 35^\circ \]