`tan ((pi)/(4) + (theta)/(2)) + tan ((pi)/(4) -(theta)/(2))` is equal to

Options

A. `sec theta`
B. `2 sec theta`
C. `sec "" (theta)/(2)`
D. `sin theta`

Answer

To solve the expression \( \tan\left(\frac{\pi}{4} + \frac{\theta}{2}\right) + \tan\left(\frac{\pi}{4} - \frac{\theta}{2}\right) \), we can use the tangent addition and subtraction formulas.

### Step 1: Use the tangent addition and subtraction formulas
The formulas for tangent of a sum and a difference are:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
\[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]

Here, let \( a = \frac{\pi}{4} \) and \( b = \frac{\theta}{2} \).

### Step 2: Calculate \( \tan\left(\frac{\pi}{4}\right) \)
We know that:
\[ \tan\left(\frac{\pi}{4}\right) = 1 \]

### Step 3: Substitute into the formulas
Substituting \( a \) and \( b \) into the formulas, we have:
\[ \tan\left(\frac{\pi}{4} + \frac{\theta}{2}\right) = \frac{1 + \tan\left(\frac{\theta}{2}\right)}{1 - 1 \cdot \tan\left(\frac{\theta}{2}\right)} = \frac{1 + \tan\left(\frac{\theta}{2}\right)}{1 - \tan\left(\frac{\theta}{2}\right)} \]
\[ \tan\left(\frac{\pi}{4} - \frac{\theta}{2}\right) = \frac{1 - \tan\left(\frac{\theta}{2}\right)}{1 + 1 \cdot \tan\left(\frac{\theta}{2}\right)} = \frac{1 - \tan\left(\frac{\theta}{2}\right)}{1 + \tan\left(\frac{\theta}{2}\right)} \]

### Step 4: Add the two tangent expressions
Now we add the two expressions:
\[ \tan\left(\frac{\pi}{4} + \frac{\theta}{2}\right) + \tan\left(\frac{\pi}{4} - \frac{\theta}{2}\right) = \frac{1 + \tan\left(\frac{\theta}{2}\right)}{1 - \tan\left(\frac{\theta}{2}\right)} + \frac{1 - \tan\left(\frac{\theta}{2}\right)}{1 + \tan\left(\frac{\theta}{2}\right)} \]

### Step 5: Find a common denominator
The common denominator for the two fractions is:
\[ (1 - \tan\left(\frac{\theta}{2}\right))(1 + \tan\left(\frac{\theta}{2}\right)) = 1 - \tan^2\left(\frac{\theta}{2}\right) \]

### Step 6: Combine the fractions
Now we can combine the fractions:
\[ = \frac{(1 + \tan\left(\frac{\theta}{2}\right))(1 + \tan\left(\frac{\theta}{2}\right)) + (1 - \tan\left(\frac{\theta}{2}\right))(1 - \tan\left(\frac{\theta}{2}\right))}{1 - \tan^2\left(\frac{\theta}{2}\right)} \]

### Step 7: Simplify the numerator
Expanding the numerator:
\[ = \frac{(1 + 2\tan\left(\frac{\theta}{2}\right) + \tan^2\left(\frac{\theta}{2}\right)) + (1 - 2\tan\left(\frac{\theta}{2}\right) + \tan^2\left(\frac{\theta}{2}\right))}{1 - \tan^2\left(\frac{\theta}{2}\right)} \]
\[ = \frac{2 + 2\tan^2\left(\frac{\theta}{2}\right)}{1 - \tan^2\left(\frac{\theta}{2}\right)} \]

### Step 8: Factor out common terms
This can be factored as:
\[ = \frac{2(1 + \tan^2\left(\frac{\theta}{2}\right))}{1 - \tan^2\left(\frac{\theta}{2}\right)} \]

### Step 9: Use the identity for tangent
Using the identity \( 1 + \tan^2 x = \sec^2 x \):
\[ = \frac{2 \sec^2\left(\frac{\theta}{2}\right)}{1 - \tan^2\left(\frac{\theta}{2}\right)} \]

### Step 10: Final result
Using the identity \( \sec^2 x = \frac{1}{\cos^2 x} \) and knowing that \( 1 - \tan^2 x = \cos 2x \):
\[ = \frac{2}{\cos\theta} \]

Thus, the final result is:
\[ \tan\left(\frac{\pi}{4} + \frac{\theta}{2}\right) + \tan\left(\frac{\pi}{4} - \frac{\theta}{2}\right) = \frac{2}{\cos\theta} \]