What is the value of `tan 15^(@).tan195^(@)`?

Options

A. `7-4sqrt(3)`
B. `7+4sqrt(3)`
C. `7+2sqrt(3)`
D. `7+6sqrt(3)`

Explanation

To find the value of \( \tan 15^\circ \cdot \tan 195^\circ \), we can use the properties of the tangent function and the angle addition formulas. Here’s a step-by-step solution:

### Step 1: Rewrite \( \tan 195^\circ \)
We know that:
\[ \tan(195^\circ) = \tan(180^\circ + 15^\circ) \]
Using the property of tangent:
\[ \tan(180^\circ + \theta) = \tan(\theta) \]
Thus:
\[ \tan(195^\circ) = \tan(15^\circ) \]

### Step 2: Substitute into the expression
Now we can substitute this back into our original expression:
\[ \tan(15^\circ) \cdot \tan(195^\circ) = \tan(15^\circ) \cdot \tan(15^\circ) = \tan^2(15^\circ) \]

### Step 3: Calculate \( \tan^2(15^\circ) \)
To find \( \tan^2(15^\circ) \), we can use the known value of \( \tan(15^\circ) \):
\[ \tan(15^\circ) = 2 - \sqrt{3} \]
Now we square this value:
\[ \tan^2(15^\circ) = (2 - \sqrt{3})^2 \]

### Step 4: Expand the square
Using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[ \tan^2(15^\circ) = 2^2 - 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 \]
\[ = 4 - 4\sqrt{3} + 3 \]
\[ = 7 - 4\sqrt{3} \]

### Step 5: Final answer
Thus, the value of \( \tan 15^\circ \cdot \tan 195^\circ \) is:
\[ \boxed{7 - 4\sqrt{3}} \]

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