`(cos 17^(@) + sin 17^(@))/(cos 17^(@) - sin 17^(@))=`

Options

A. `tan 62^(@)`
B. `tan 56^(@)`
C. `tan 54^(@)`
D. `tan73^(@)`

Answer

To solve the expression \((\cos 17^\circ + \sin 17^\circ)/(\cos 17^\circ - \sin 17^\circ)\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{\cos 17^\circ + \sin 17^\circ}{\cos 17^\circ - \sin 17^\circ} \]

### Step 2: Multiply numerator and denominator by \(\sqrt{2}\)
To simplify the expression, we can multiply both the numerator and the denominator by \(\sqrt{2}\):
\[ \frac{\sqrt{2}(\cos 17^\circ + \sin 17^\circ)}{\sqrt{2}(\cos 17^\circ - \sin 17^\circ)} \]

### Step 3: Use the angle addition formulas
Using the angle addition formulas, we know:
\[ \sqrt{2}(\cos 17^\circ + \sin 17^\circ) = \sqrt{2} \cdot \cos 17^\circ + \sqrt{2} \cdot \sin 17^\circ = \sqrt{2} \cdot \sin(17^\circ + 45^\circ) = \sqrt{2} \cdot \sin 62^\circ \]
and
\[ \sqrt{2}(\cos 17^\circ - \sin 17^\circ) = \sqrt{2} \cdot \cos 17^\circ - \sqrt{2} \cdot \sin 17^\circ = \sqrt{2} \cdot \cos(17^\circ + 45^\circ) = \sqrt{2} \cdot \cos 62^\circ \]

### Step 4: Substitute back into the expression
Now we substitute these results back into our expression:
\[ \frac{\sqrt{2} \cdot \sin 62^\circ}{\sqrt{2} \cdot \cos 62^\circ} \]

### Step 5: Simplify the expression
The \(\sqrt{2}\) cancels out:
\[ \frac{\sin 62^\circ}{\cos 62^\circ} = \tan 62^\circ \]

### Final Answer
Thus, the final result is:
\[ \tan 62^\circ \]