`(cos 9^(@) + sin 9^(@))/(cos 9^(@) - sin 9^(@))` equals

Options

A. `tan 26^(@)`
B. `tan 81^(@)`
C. `tan 51^(@)`
D. `tan 54^(@)`

Answer

To solve the expression \((\cos 9^\circ + \sin 9^\circ) / (\cos 9^\circ - \sin 9^\circ)\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the original expression:
\[ \frac{\cos 9^\circ + \sin 9^\circ}{\cos 9^\circ - \sin 9^\circ} \]

### Step 2: Factor out common terms
Notice that we can express the numerator and denominator in terms of tangent:
\[ \frac{\cos 9^\circ + \sin 9^\circ}{\cos 9^\circ - \sin 9^\circ} = \frac{1 + \tan 9^\circ}{1 - \tan 9^\circ} \]
where \(\tan 9^\circ = \frac{\sin 9^\circ}{\cos 9^\circ}\).

### Step 3: Use the tangent addition formula
We can use the tangent addition formula, which states:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
In our case, let \(A = 45^\circ\) and \(B = 9^\circ\):
\[ \tan(45^\circ + 9^\circ) = \frac{\tan 45^\circ + \tan 9^\circ}{1 - \tan 45^\circ \tan 9^\circ} \]
Since \(\tan 45^\circ = 1\), we have:
\[ \tan(54^\circ) = \frac{1 + \tan 9^\circ}{1 - \tan 9^\circ} \]

### Step 4: Conclude the result
Thus, we can conclude that:
\[ \frac{\cos 9^\circ + \sin 9^\circ}{\cos 9^\circ - \sin 9^\circ} = \tan(54^\circ) \]

### Final Answer
The expression \((\cos 9^\circ + \sin 9^\circ) / (\cos 9^\circ - \sin 9^\circ)\) equals \(\tan(54^\circ)\).

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