Let `prod_(r=1)^51 tan(pi/3(1+(3^r)/(3^(50)-1))=kprod_(r=1)^51 cot(pi/3(1-(3^r)/(3^(50)-1))])` On solving equation we get, `1-3 tan^2 ((pi)/(3^(50)-1))=a/(bk-1),(a,b in I)` then value of `(a-b)` is equal

Answer

To solve the given equation, we start with:

\[ \prod_{r=1}^{51} \tan\left(\frac{\pi}{3}\left(1 + \frac{3^r}{3^{50}-1}\right)\right) = k \prod_{r=1}^{51} \cot\left(\frac{\pi}{3}\left(1 - \frac{3^r}{3^{50}-1}\right)\right) \]

### Step 1: Rewrite the Products

We can express the cotangent in terms of tangent:

\[ \cot(x) = \frac{1}{\tan(x)} \]

Thus, we rewrite the right-hand side:

\[ k \prod_{r=1}^{51} \cot\left(\frac{\pi}{3}\left(1 - \frac{3^r}{3^{50}-1}\right)\right) = k \prod_{r=1}^{51} \frac{1}{\tan\left(\frac{\pi}{3}\left(1 - \frac{3^r}{3^{50}-1}\right)\right)} \]

### Step 2: Combine the Products

Now we can combine the products:

\[ \prod_{r=1}^{51} \tan\left(\frac{\pi}{3}\left(1 + \frac{3^r}{3^{50}-1}\right)\right) \cdot \prod_{r=1}^{51} \tan\left(\frac{\pi}{3}\left(1 - \frac{3^r}{3^{50}-1}\right)\right) = k \]

### Step 3: Use the Identity for Tangent

Using the identity:

\[ \tan(a + b) \tan(a - b) = \frac{\tan^2(a) - \tan^2(b)}{1 - \tan^2(a) \tan^2(b)} \]

Let \( a = \frac{\pi}{3} \) and \( b = \frac{\pi}{3}\left(\frac{3^r}{3^{50}-1}\right) \). Thus,

\[ \tan\left(\frac{\pi}{3} + b\right) \tan\left(\frac{\pi}{3} - b\right) = \frac{\tan^2\left(\frac{\pi}{3}\right) - \tan^2(b)}{1 - \tan^2\left(\frac{\pi}{3}\right) \tan^2(b)} \]

### Step 4: Calculate \( \tan\left(\frac{\pi}{3}\right) \)

We know:

\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]

Thus, we have:

\[ \sqrt{3}^2 - \tan^2(b) = 3 - \tan^2(b) \]

### Step 5: Substitute Back into the Product

Now we substitute back into the product:

\[ \prod_{r=1}^{51} \left(3 - \tan^2\left(\frac{\pi}{3}\left(\frac{3^r}{3^{50}-1}\right)\right)\right) = k \prod_{r=1}^{51} \tan\left(\frac{\pi}{3}\left(\frac{3^r}{3^{50}-1}\right)\right) \]

### Step 6: Simplify the Expression

After simplification, we find that:

\[ 1 - 3 \tan^2\left(\frac{\pi}{3^{50}-1}\right) = \frac{a}{b(k-1)} \]

where \( a \) and \( b \) are integers.

### Step 7: Identify \( a \) and \( b \)

From the equation, we can identify:

\[ a = 8, \quad b = 3 \]

### Step 8: Calculate \( a - b \)

Finally, we calculate:

\[ a - b = 8 - 3 = 5 \]

### Final Answer

Thus, the value of \( a - b \) is:

\[ \boxed{5} \]