If `tan ((pi)/(4) + theta) + tan ((pi )/(4) - theta) = p sec 2 theta)` then the vlaue of p is equal to :

Options

A. 2
B. 3
C. 1
D. 4

Answer

To solve the equation \( \tan\left(\frac{\pi}{4} + \theta\right) + \tan\left(\frac{\pi}{4} - \theta\right) = p \sec^2 \theta \), we will follow these steps:

### Step 1: Use the tangent addition and subtraction formulas
We know that:
\[ \tan\left(\frac{\pi}{4} + \theta\right) = \frac{\tan\frac{\pi}{4} + \tan\theta}{1 - \tan\frac{\pi}{4} \tan\theta} \]
and
\[ \tan\left(\frac{\pi}{4} - \theta\right) = \frac{\tan\frac{\pi}{4} - \tan\theta}{1 + \tan\frac{\pi}{4} \tan\theta} \]
Since \( \tan\frac{\pi}{4} = 1 \), we can substitute this into our expressions:
\[ \tan\left(\frac{\pi}{4} + \theta\right) = \frac{1 + \tan\theta}{1 - \tan\theta} \]
\[ \tan\left(\frac{\pi}{4} - \theta\right) = \frac{1 - \tan\theta}{1 + \tan\theta} \]

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

### Step 3: Find a common denominator
The common denominator for the two fractions is \( (1 - \tan\theta)(1 + \tan\theta) \):
\[ = \frac{(1 + \tan\theta)^2 + (1 - \tan\theta)^2}{(1 - \tan\theta)(1 + \tan\theta)} \]

### Step 4: Simplify the numerator
Expanding the numerator:
\[ (1 + \tan\theta)^2 = 1 + 2\tan\theta + \tan^2\theta \]
\[ (1 - \tan\theta)^2 = 1 - 2\tan\theta + \tan^2\theta \]
Adding these:
\[ 1 + 2\tan\theta + \tan^2\theta + 1 - 2\tan\theta + \tan^2\theta = 2 + 2\tan^2\theta \]

### Step 5: Write the complete expression
Now we have:
\[ \tan\left(\frac{\pi}{4} + \theta\right) + \tan\left(\frac{\pi}{4} - \theta\right) = \frac{2 + 2\tan^2\theta}{1 - \tan^2\theta} \]
Factoring out the 2 from the numerator gives:
\[ = \frac{2(1 + \tan^2\theta)}{1 - \tan^2\theta} \]

### Step 6: Use the identity for secant
Recall that \( 1 + \tan^2\theta = \sec^2\theta \):
\[ = \frac{2 \sec^2\theta}{1 - \tan^2\theta} \]

### Step 7: Relate to the original equation
Now we need to relate this to \( p \sec^2\theta \):
\[ \frac{2 \sec^2\theta}{1 - \tan^2\theta} = p \sec^2\theta \]
Dividing both sides by \( \sec^2\theta \) (assuming \( \sec^2\theta \neq 0 \)):
\[ \frac{2}{1 - \tan^2\theta} = p \]

### Step 8: Identify the value of \( p \)
From the equation, we can see that \( p = 2 \).

Thus, the value of \( p \) is:
\[ \boxed{2} \]