`tan((pi)/(4)+theta)-tan((pi)/(4)-theta)=?`

Options

A. `2tan2theta`
B. `2cot2theta`
C. `tan2theta`
D. `cot2theta`

Answer

To solve the problem \( \tan\left(\frac{\pi}{4} + \theta\right) - \tan\left(\frac{\pi}{4} - \theta\right) \), we can use the tangent addition and subtraction formulas.

### Step-by-Step Solution:

1. **Identify the formulas**:
We will use the tangent addition and subtraction formulas:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
Here, let \( A = \frac{\pi}{4} \) and \( B = \theta \).

2. **Calculate \( \tan\left(\frac{\pi}{4} + \theta\right) \)**:
Using the addition formula:
\[ \tan\left(\frac{\pi}{4} + \theta\right) = \frac{\tan\left(\frac{\pi}{4}\right) + \tan(\theta)}{1 - \tan\left(\frac{\pi}{4}\right) \tan(\theta)} \]
Since \( \tan\left(\frac{\pi}{4}\right) = 1 \):
\[ \tan\left(\frac{\pi}{4} + \theta\right) = \frac{1 + \tan(\theta)}{1 - 1 \cdot \tan(\theta)} = \frac{1 + \tan(\theta)}{1 - \tan(\theta)} \]

3. **Calculate \( \tan\left(\frac{\pi}{4} - \theta\right) \)**:
Using the subtraction formula:
\[ \tan\left(\frac{\pi}{4} - \theta\right) = \frac{\tan\left(\frac{\pi}{4}\right) - \tan(\theta)}{1 + \tan\left(\frac{\pi}{4}\right) \tan(\theta)} \]
Again, since \( \tan\left(\frac{\pi}{4}\right) = 1 \):
\[ \tan\left(\frac{\pi}{4} - \theta\right) = \frac{1 - \tan(\theta)}{1 + 1 \cdot \tan(\theta)} = \frac{1 - \tan(\theta)}{1 + \tan(\theta)} \]

4. **Subtract the two results**:
Now we need to compute:
\[ \tan\left(\frac{\pi}{4} + \theta\right) - \tan\left(\frac{\pi}{4} - \theta\right) \]
Substituting the values we found:
\[ = \left(\frac{1 + \tan(\theta)}{1 - \tan(\theta)}\right) - \left(\frac{1 - \tan(\theta)}{1 + \tan(\theta)}\right) \]

5. **Find a common denominator**:
The common denominator is \( (1 - \tan(\theta))(1 + \tan(\theta)) \):
\[ = \frac{(1 + \tan(\theta))^2 - (1 - \tan(\theta))^2}{(1 - \tan(\theta))(1 + \tan(\theta))} \]

6. **Simplify the numerator**:
Using the difference of squares:
\[ = \frac{(1 + \tan(\theta) + 1 - \tan(\theta))(1 + \tan(\theta) - (1 - \tan(\theta)))}{(1 - \tan(\theta))(1 + \tan(\theta))} \]
This simplifies to:
\[ = \frac{(2)(2\tan(\theta))}{(1 - \tan^2(\theta))} \]
Thus:
\[ = \frac{4\tan(\theta)}{1 - \tan^2(\theta)} \]

7. **Final Result**:
We can express this in terms of \( \tan(2\theta) \):
\[ = 2 \cdot \frac{2\tan(\theta)}{1 - \tan^2(\theta)} = 2\tan(2\theta) \]

### Final Answer:
\[ \tan\left(\frac{\pi}{4} + \theta\right) - \tan\left(\frac{\pi}{4} - \theta\right) = 2\tan(2\theta) \]