If `3tan(theta-15^0)=tan(theta+15^0)0

Explanation

To solve the equation \(3 \tan(\theta - 15^\circ) = \tan(\theta + 15^\circ)\) for \(0 < \theta < 90^\circ\), 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
Using the tangent addition and subtraction formulas:
\[ \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \]
we can express \(\tan(\theta - 15^\circ)\) and \(\tan(\theta + 15^\circ)\):
\[ \tan(\theta - 15^\circ) = \frac{\tan \theta - \tan 15^\circ}{1 + \tan \theta \tan 15^\circ} \]
\[ \tan(\theta + 15^\circ) = \frac{\tan \theta + \tan 15^\circ}{1 - \tan \theta \tan 15^\circ} \]

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

### Step 4: Cross-multiply to eliminate the fractions
Cross-multiplying yields:
\[ 3(\tan \theta - \tan 15^\circ)(1 - \tan \theta \tan 15^\circ) = (\tan \theta + \tan 15^\circ)(1 + \tan \theta \tan 15^\circ) \]

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

### Step 6: Rearranging the equation
Rearranging terms will help to isolate \(\tan \theta\):
\[ 3 \tan \theta - 3 \tan 15^\circ - 3 \tan \theta \tan 15^\circ \tan \theta + 3 \tan 15^\circ \tan^2 \theta = \tan \theta + \tan 15^\circ + \tan^2 \theta \tan 15^\circ + \tan \theta \tan^2 15^\circ \]

### Step 7: Collect like terms
Collecting like terms will allow us to simplify and solve for \(\tan \theta\).

### Step 8: Solve for \(\theta\)
Once we have \(\tan \theta\), we can find \(\theta\) using the inverse tangent function.

### Final Step: Check the value of \(\theta\)
Ensure that the value of \(\theta\) lies within the specified range \(0 < \theta < 90^\circ\).