General solution of the equation
`tan theta + tan 4 theta + tan 7 theta = tan theta cdot tan 4 theta cdot tan 7 theta "is" theta = `

Options

A. `(npi)/(12),/n in 1`
B. `(npi)/(4), n in 1`
C. `(n pi)/(3), n in 1`
D. `(npi)/(6), n in 1`

Answer

To find the general solution of the equation

\[ \tan \theta + \tan 4\theta + \tan 7\theta = \tan \theta \cdot \tan 4\theta \cdot \tan 7\theta, \]

we can use the identity for the tangent of the sum of angles. The identity states that:

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

### Step 1: Identify the angles
Let \( A = \theta \), \( B = 4\theta \), and \( C = 7\theta \). Then we can rewrite the equation as:

\[ \tan(\theta + 4\theta + 7\theta) = \tan(180^\circ). \]

### Step 2: Sum the angles
Now we sum the angles:

\[ \theta + 4\theta + 7\theta = 12\theta. \]

### Step 3: Set the equation
Since we know that \( \tan(180^\circ) = 0 \), we can set up the equation:

\[ \tan(12\theta) = 0. \]

### Step 4: Solve for \( \theta \)
The general solution for \( \tan x = 0 \) is given by:

\[ x = n\pi \quad (n \in \mathbb{Z}). \]

So we have:

\[ 12\theta = n\pi. \]

Dividing both sides by 12 gives:

\[ \theta = \frac{n\pi}{12}. \]

### Step 5: Convert to degrees
If we want the solution in degrees, we can convert:

\[ \theta = \frac{n \cdot 180^\circ}{12} = \frac{15n}{1}^\circ. \]

### Final General Solution
Thus, the general solution for \( \theta \) is:

\[ \theta = 15n^\circ \quad (n \in \mathbb{Z}). \]

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