If A=`tan^-1((xsqrt3)/(2k-x))` and B=`tan^-1((2x-k)/(ksqrt3))` then find the value of `A-B` (A) `0^@` (B) `30^@` (C) `60^@` (D) `45^@`

Options

A. `0^(@)`
B. `45^(@)`
C. `60^(@)`
D. `30^(@)`

Answer

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

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

### Step 1: Use the formula for the difference of inverse tangents

We can use the formula for the difference of two inverse tangents:

\[ \tan^{-1}(x) - \tan^{-1}(y) = \tan^{-1}\left(\frac{x - y}{1 + xy}\right) \]

In our case, let:
- \( x = \frac{x \sqrt{3}}{2k - x} \)
- \( y = \frac{2x - k}{k \sqrt{3}} \)

Thus, we can write:

\[ A - B = \tan^{-1}\left(\frac{x \sqrt{3}}{2k - x} - \frac{2x - k}{k \sqrt{3}} \bigg/ \left(1 + \frac{x \sqrt{3}}{2k - x} \cdot \frac{2x - k}{k \sqrt{3}}\right)\right) \]

### Step 2: Simplify the numerator

The numerator becomes:

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

Expanding the numerator:

\[ = \frac{3kx - (4kx - 2k^2 - 2x^2 + kx)}{(2k - x)(k \sqrt{3})} \]
\[ = \frac{3kx - 4kx + 2k^2 + 2x^2 - kx}{(2k - x)(k \sqrt{3})} \]
\[ = \frac{-2kx + 2k^2 + 2x^2}{(2k - x)(k \sqrt{3})} \]

### Step 3: Simplify the denominator

Now, we simplify the denominator:

\[ 1 + \frac{x \sqrt{3}}{2k - x} \cdot \frac{2x - k}{k \sqrt{3}} = 1 + \frac{x(2x - k)}{(2k - x)k} \]
\[ = \frac{(2k - x)k + x(2x - k)}{(2k - x)k} \]
\[ = \frac{2k^2 - kx + 2x^2 - kx}{(2k - x)k} \]
\[ = \frac{2k^2 + 2x^2 - 2kx}{(2k - x)k} \]

### Step 4: Combine the results

Now we can combine the numerator and denominator:

\[ A - B = \tan^{-1}\left(\frac{-2kx + 2k^2 + 2x^2}{2k^2 + 2x^2 - 2kx}\right) \]

### Step 5: Factor out common terms

Both the numerator and denominator can be simplified:

\[ = \tan^{-1}\left(\frac{2(k^2 + x^2 - kx)}{2(k^2 + x^2 - kx)}\right) = \tan^{-1}(1) \]

### Step 6: Find the angle

Since \( \tan^{-1}(1) = 45^\circ \):

Thus, we conclude that:

\[ A - B = 45^\circ \]

### Final Answer

The value of \( A - B \) is \( 45^\circ \).