If `( cos 13^(@)+sin13^(@))/(cos13^(@)-sin13^(@))=tan A`, then A=_________

Answer

To solve the equation

\[ \frac{\cos 13^\circ + \sin 13^\circ}{\cos 13^\circ - \sin 13^\circ} = \tan A, \]

we will follow these steps:

### Step 1: Divide numerator and denominator by \(\cos 13^\circ\)

We can simplify the expression by dividing both the numerator and the denominator by \(\cos 13^\circ\):

\[ \frac{\frac{\cos 13^\circ}{\cos 13^\circ} + \frac{\sin 13^\circ}{\cos 13^\circ}}{\frac{\cos 13^\circ}{\cos 13^\circ} - \frac{\sin 13^\circ}{\cos 13^\circ}} = \frac{1 + \tan 13^\circ}{1 - \tan 13^\circ}. \]

### Step 2: Recognize the formula

The expression \(\frac{1 + \tan 13^\circ}{1 - \tan 13^\circ}\) can be recognized as the tangent addition formula:

\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}. \]

In our case, we can set \(A = 45^\circ\) and \(B = 13^\circ\):

\[ \tan(45^\circ + 13^\circ) = \frac{\tan 45^\circ + \tan 13^\circ}{1 - \tan 45^\circ \tan 13^\circ}. \]

### Step 3: Substitute values

Since \(\tan 45^\circ = 1\), we can substitute this into our equation:

\[ \tan(45^\circ + 13^\circ) = \frac{1 + \tan 13^\circ}{1 - \tan 13^\circ}. \]

### Step 4: Solve for \(A\)

From the above, we have:

\[ \tan A = \tan(45^\circ + 13^\circ). \]

Thus,

\[ A = 45^\circ + 13^\circ = 58^\circ. \]

### Final Answer

Therefore, the value of \(A\) is

\[ \boxed{58^\circ}. \]

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