`" tan " ((pi)/(4) -x) =(1 - tan x)/( 1+ tan x)`

Options

A. `tan (x -(pi)/(4))`
B. `tan (x + (pi)/(4))`
C. `tan ((pi)/(4) -x)`
D. `tan((pi)/(4)) - tan(x)`

Answer

To solve the equation \( \tan\left(\frac{\pi}{4} - x\right) = \frac{1 - \tan x}{1 + \tan x} \), we will start with the left-hand side (LHS) and simplify it using trigonometric identities.

### Step 1: Write down the left-hand side
We start with:
\[ \text{LHS} = \tan\left(\frac{\pi}{4} - x\right) \]

### Step 2: Use the tangent subtraction formula
The tangent subtraction formula states:
\[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]
In our case, let \( a = \frac{\pi}{4} \) and \( b = x \). We know that \( \tan\left(\frac{\pi}{4}\right) = 1 \). Thus, we can substitute into the formula:
\[ \tan\left(\frac{\pi}{4} - x\right) = \frac{\tan\left(\frac{\pi}{4}\right) - \tan x}{1 + \tan\left(\frac{\pi}{4}\right) \tan x} = \frac{1 - \tan x}{1 + 1 \cdot \tan x} = \frac{1 - \tan x}{1 + \tan x} \]

### Step 3: Write down the right-hand side
Now, we can write down the right-hand side (RHS):
\[ \text{RHS} = \frac{1 - \tan x}{1 + \tan x} \]

### Step 4: Compare LHS and RHS
Now we see that:
\[ \text{LHS} = \frac{1 - \tan x}{1 + \tan x} = \text{RHS} \]

### Conclusion
Since both sides are equal, we have proved that:
\[ \tan\left(\frac{\pi}{4} - x\right) = \frac{1 - \tan x}{1 + \tan x} \]