What is the value of `(1+tan15^(@))/(1-tan15^(@))`?

Options

A. 1
B. `(1)/(sqrt(2))`
C. `(1)/(sqrt(3))`
D. `sqrt(3)`

Explanation

To find the value of \(\frac{1 + \tan 15^\circ}{1 - \tan 15^\circ}\), we can use the tangent addition formula. Let's go through the solution step by step.

### Step 1: Write the expression
We start with the expression:
\[ \frac{1 + \tan 15^\circ}{1 - \tan 15^\circ} \]

### Step 2: 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, we can set \(A = 45^\circ\) and \(B = 15^\circ\). Therefore, we have:
\[ \tan(45^\circ + 15^\circ) = \tan 60^\circ \]
This gives us:
\[ \tan 60^\circ = \frac{\tan 45^\circ + \tan 15^\circ}{1 - \tan 45^\circ \tan 15^\circ} \]

### Step 3: Substitute known values
We know that:
\[ \tan 45^\circ = 1 \quad \text{and} \quad \tan 60^\circ = \sqrt{3} \]
Substituting these values into the formula, we get:
\[ \sqrt{3} = \frac{1 + \tan 15^\circ}{1 - \tan 15^\circ} \]

### Step 4: Conclusion
Thus, we conclude that:
\[ \frac{1 + \tan 15^\circ}{1 - \tan 15^\circ} = \sqrt{3} \]

### Final Answer
The value of \(\frac{1 + \tan 15^\circ}{1 - \tan 15^\circ}\) is \(\sqrt{3}\).

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