`(cos9^(@)+sin9^(@))/(cos9^(@)-sin9^(@))=?`

Options

A. `tan54^(@)`
B. `tan36^(@)`
C. `tan18^(@)`
D. `cot18^(@)`

Answer

To solve the expression \((\cos 9^\circ + \sin 9^\circ) / (\cos 9^\circ - \sin 9^\circ)\), we can use a known trigonometric identity.

### Step-by-Step Solution:

1. **Identify the Expression**:
We have the expression \(\frac{\cos 9^\circ + \sin 9^\circ}{\cos 9^\circ - \sin 9^\circ}\).

2. **Use the Trigonometric Identity**:
The expression \(\frac{\cos x + \sin x}{\cos x - \sin x}\) can be simplified using the identity:
\[ \frac{\cos x + \sin x}{\cos x - \sin x} = \tan\left(x + 45^\circ\right) \]
Here, \(x = 9^\circ\).

3. **Apply the Identity**:
Substitute \(x\) into the identity:
\[ \frac{\cos 9^\circ + \sin 9^\circ}{\cos 9^\circ - \sin 9^\circ} = \tan\left(9^\circ + 45^\circ\right) \]

4. **Calculate the Angle**:
Now calculate \(9^\circ + 45^\circ\):
\[ 9^\circ + 45^\circ = 54^\circ \]

5. **Final Expression**:
Therefore, we have:
\[ \frac{\cos 9^\circ + \sin 9^\circ}{\cos 9^\circ - \sin 9^\circ} = \tan(54^\circ) \]

### Conclusion:
The final answer is:
\[ \tan(54^\circ) \]