`(cos17^(@)+sin17^(@))/(cos17^(@)-sin17^(@))=?`

Options

A. `tan62^(@)`
B. `tan56^(@)`
C. `tan54^(@)`
D. `tan74^(@)`

Answer

To solve the expression \((\cos 17^\circ + \sin 17^\circ) / (\cos 17^\circ - \sin 17^\circ)\), we can use a trigonometric identity.

### Step-by-step Solution:

1. **Identify the Expression**:
We have the expression:
\[ \frac{\cos 17^\circ + \sin 17^\circ}{\cos 17^\circ - \sin 17^\circ} \]

2. **Use the Trigonometric Identity**:
The expression \(\frac{\cos x + \sin x}{\cos x - \sin x}\) can be simplified using the identity:
\[ \frac{\cos x + \sin x}{\cos x - \sin x} = \tan\left(x + \frac{\pi}{4}\right) \]
Here, \(x = 17^\circ\).

3. **Apply the Identity**:
Substitute \(x\) into the identity:
\[ \frac{\cos 17^\circ + \sin 17^\circ}{\cos 17^\circ - \sin 17^\circ} = \tan\left(17^\circ + 45^\circ\right) \]

4. **Calculate the Angle**:
Now, calculate \(17^\circ + 45^\circ\):
\[ 17^\circ + 45^\circ = 62^\circ \]

5. **Final Result**:
Therefore, we have:
\[ \frac{\cos 17^\circ + \sin 17^\circ}{\cos 17^\circ - \sin 17^\circ} = \tan(62^\circ) \]

### Conclusion:
The value of \(\frac{\cos 17^\circ + \sin 17^\circ}{\cos 17^\circ - \sin 17^\circ}\) is \(\tan(62^\circ)\).

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