If `3 tan (theta-15^(@))=tan(theta+15^(@)), 0lt thetalt pi,then theta=`

Options

A. `(pi)/(2)`
B. `(pi)/(4)`
C. `(3pi)/(4)`
D. `(pi)/(6)`

Answer

To solve the equation \( 3 \tan(\theta - 15^\circ) = \tan(\theta + 15^\circ) \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ 3 \tan(\theta - 15^\circ) = \tan(\theta + 15^\circ) \]

### Step 2: Use the tangent addition and subtraction formulas
Recall the tangent addition and subtraction formulas:
\[ \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \]
Let \( A = \theta \) and \( B = 15^\circ \). Therefore, we can express:
\[ \tan(\theta + 15^\circ) = \frac{\tan \theta + \tan 15^\circ}{1 - \tan \theta \tan 15^\circ} \]
\[ \tan(\theta - 15^\circ) = \frac{\tan \theta - \tan 15^\circ}{1 + \tan \theta \tan 15^\circ} \]

### Step 3: Substitute the formulas into the equation
Substituting these into the equation gives:
\[ 3 \cdot \frac{\tan \theta - \tan 15^\circ}{1 + \tan \theta \tan 15^\circ} = \frac{\tan \theta + \tan 15^\circ}{1 - \tan \theta \tan 15^\circ} \]

### Step 4: Cross-multiply
Cross-multiplying results in:
\[ 3(\tan \theta - \tan 15^\circ)(1 - \tan \theta \tan 15^\circ) = (\tan \theta + \tan 15^\circ)(1 + \tan \theta \tan 15^\circ) \]

### Step 5: Expand both sides
Expanding both sides:
\[ 3(\tan \theta - \tan 15^\circ - \tan \theta \tan 15^\circ \tan \theta + \tan 15^\circ \tan^2 \theta) = \tan \theta + \tan 15^\circ + \tan \theta \tan^2 15^\circ + \tan^2 \theta \tan 15^\circ \]

### Step 6: Rearranging the equation
Rearranging the equation to isolate terms involving \( \tan \theta \) gives us a polynomial in \( \tan \theta \).

### Step 7: Solve for \( \tan \theta \)
This will lead to a quadratic equation in terms of \( \tan \theta \). Solving this quadratic equation will yield the possible values for \( \tan \theta \).

### Step 8: Find \( \theta \)
Once we have the values for \( \tan \theta \), we can find \( \theta \) by taking the arctangent and ensuring that \( 0 < \theta < \pi \).

### Final Answer
After solving the quadratic equation and applying the arctangent, we will find the value(s) of \( \theta \) that satisfy the original equation.

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