If `phi= tan^(-1)(( xsqrt3)/(2k-x))` and `psi = tan^(-1) ((2x-k)/(k*sqrt3))` then one value of `phi - psi` is

Options

A. `pi/6`
B. `pi/3`
C. `(5pi)/6`
D. `(4pi)/3`

Answer

To solve the problem, we need to find the value of \( \phi - \psi \) where:

\[ \phi = \tan^{-1}\left(\frac{x\sqrt{3}}{2k - x}\right) \]
\[ \psi = \tan^{-1}\left(\frac{2x - k}{k\sqrt{3}}\right) \]

We will use the formula for the difference of inverse tangents:

\[ \tan^{-1} A - \tan^{-1} B = \tan^{-1}\left(\frac{A - B}{1 + AB}\right) \]

### Step 1: Identify \( A \) and \( B \)

Let:
\[ A = \frac{x\sqrt{3}}{2k - x} \]
\[ B = \frac{2x - k}{k\sqrt{3}} \]

### Step 2: Calculate \( A - B \)

Now, we calculate \( A - B \):

\[ A - B = \frac{x\sqrt{3}}{2k - x} - \frac{2x - k}{k\sqrt{3}} \]

To subtract these fractions, we need a common denominator:

\[ A - B = \frac{x\sqrt{3} \cdot k\sqrt{3} - (2x - k)(2k - x)}{(2k - x)(k\sqrt{3})} \]

### Step 3: Expand the numerator

Expanding the numerator:

1. First term:
\[ x\sqrt{3} \cdot k\sqrt{3} = 3kx \]

2. Second term:
\[ (2x - k)(2k - x) = 4kx - 2x^2 - 2k^2 + kx \]
So, the second term becomes:
\[ - (4kx - 2x^2 - 2k^2 + kx) = -4kx + 2x^2 + 2k^2 - kx \]

Combining these gives:

\[ 3kx - (4kx - 2x^2 + 2k^2 - kx) = 3kx - 4kx + 2x^2 + 2k^2 - kx = -2kx + 2x^2 + 2k^2 \]

### Step 4: Calculate \( 1 + AB \)

Next, we calculate \( 1 + AB \):

\[ AB = \left(\frac{x\sqrt{3}}{2k - x}\right) \left(\frac{2x - k}{k\sqrt{3}}\right) = \frac{x(2x - k)}{(2k - x)k} \]

So,

\[ 1 + AB = 1 + \frac{x(2x - k)}{(2k - x)k} = \frac{(2k - x)k + x(2x - k)}{(2k - x)k} \]

### Step 5: Combine results

Now we can substitute \( A - B \) and \( 1 + AB \) into the formula:

\[ \phi - \psi = \tan^{-1}\left(\frac{A - B}{1 + AB}\right) \]

### Step 6: Simplify

After simplifying the numerator and denominator, we find that:

\[ \phi - \psi = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) \]

### Step 7: Final Result

We know that:

\[ \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \]

Thus, one value of \( \phi - \psi \) is:

\[ \phi - \psi = \frac{\pi}{6} \]