If tan `69^(@) + tan 66^(@) - tan 69^(@)tan 66^(@) = 2k ` then k = (a)`-(1)/(2)` (b)`(1)/(2)` (c)`-1` (d)None of these

Options

A. `-(1)/(2)`
B. `(1)/(2)`
C. `-1`
D. None of these

Answer

To solve the problem, we start with the equation given in the question:

\[ \tan 69^\circ + \tan 66^\circ - \tan 69^\circ \tan 66^\circ = 2k \]

### Step 1: Use the angle addition formula for tangent
We know that:

\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

In this case, we can set \(A = 69^\circ\) and \(B = 66^\circ\). Therefore, we can write:

\[ \tan(69^\circ + 66^\circ) = \tan 135^\circ \]

### Step 2: Calculate \(\tan 135^\circ\)
We know that:

\[ \tan 135^\circ = -\tan 45^\circ = -1 \]

### Step 3: Substitute into the equation
Now substituting this back into the equation we derived from the angle addition formula:

\[ \tan 135^\circ = \frac{\tan 69^\circ + \tan 66^\circ}{1 - \tan 69^\circ \tan 66^\circ} \]

This gives us:

\[ -1 = \frac{\tan 69^\circ + \tan 66^\circ}{1 - \tan 69^\circ \tan 66^\circ} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives:

\[ - (1 - \tan 69^\circ \tan 66^\circ) = \tan 69^\circ + \tan 66^\circ \]

### Step 5: Rearranging the equation
This can be rearranged to:

\[ -\tan 69^\circ - \tan 66^\circ = -1 + \tan 69^\circ \tan 66^\circ \]

### Step 6: Simplifying the equation
This simplifies to:

\[ \tan 69^\circ + \tan 66^\circ - \tan 69^\circ \tan 66^\circ = 1 \]

### Step 7: Relate to the original equation
From the original equation, we have:

\[ \tan 69^\circ + \tan 66^\circ - \tan 69^\circ \tan 66^\circ = 2k \]

Thus, we can equate:

\[ 2k = 1 \]

### Step 8: Solve for \(k\)
Dividing both sides by 2 gives:

\[ k = \frac{1}{2} \]

### Conclusion
The value of \(k\) is:

\[ \boxed{\frac{1}{2}} \]