If `tan( pi/4 + x). tan(pi/4 - x) = 1` , then x equals

Options

A. `pi/6`
B. `pi/3`
C. `pi/4`
D. Zero

Answer

To solve the equation \( \tan\left(\frac{\pi}{4} + x\right) \cdot \tan\left(\frac{\pi}{4} - x\right) = 1 \), we can follow these steps:

### Step 1: Use the tangent addition and subtraction formulas
We know that:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
\[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]
In our case, let \( a = \frac{\pi}{4} \) and \( b = x \).

### Step 2: Substitute \( a \) and \( b \) into the formulas
We have:
\[ \tan\left(\frac{\pi}{4} + x\right) = \frac{\tan\frac{\pi}{4} + \tan x}{1 - \tan\frac{\pi}{4} \tan x} = \frac{1 + \tan x}{1 - \tan x} \]
\[ \tan\left(\frac{\pi}{4} - x\right) = \frac{\tan\frac{\pi}{4} - \tan x}{1 + \tan\frac{\pi}{4} \tan x} = \frac{1 - \tan x}{1 + \tan x} \]

### Step 3: Multiply the two tangent expressions
Now, we multiply these two results:
\[ \tan\left(\frac{\pi}{4} + x\right) \cdot \tan\left(\frac{\pi}{4} - x\right) = \left(\frac{1 + \tan x}{1 - \tan x}\right) \cdot \left(\frac{1 - \tan x}{1 + \tan x}\right) \]

### Step 4: Simplify the expression
The expression simplifies to:
\[ = \frac{(1 + \tan x)(1 - \tan x)}{(1 - \tan x)(1 + \tan x)} = 1 \]
This confirms that the equation holds true for any value of \( x \).

### Step 5: Conclusion
Since the equation \( \tan\left(\frac{\pi}{4} + x\right) \cdot \tan\left(\frac{\pi}{4} - x\right) = 1 \) is satisfied for all \( x \), we conclude that \( x \) can take any value.

### Final Answer
Thus, \( x \) can be any real number.

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