Statement -1: If `(pi)/(12)lethetale(pi)/(3),` then
`sin(theta-(pi)/(4))sin(theta-(7pi)/(12))sin(theta+(pi)/(12))"lies between"-(1)/(4sqrt2)and 1/4.`
Statement-2: The value of `sin thetasin((pi)/(3)-theta)sin((pi)/(3)+theta)is 1/4sin3theta.`

Options

A. Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for Statement -1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Explanation

To solve the problem, we need to analyze both statements and derive the necessary conclusions step by step.

### Step 1: Analyzing Statement 2

We start with Statement 2, which states that:
\[ \sin(\theta) \sin\left(\frac{\pi}{3} - \theta\right) \sin\left(\frac{\pi}{3} + \theta\right) = \frac{1}{4} \sin(3\theta) \]

Using the sine addition and subtraction formulas, we can rewrite the left-hand side:
\[ \sin\left(\frac{\pi}{3} - \theta\right) = \sin\left(\frac{\pi}{3}\right)\cos(\theta) - \cos\left(\frac{\pi}{3}\right)\sin(\theta) = \frac{\sqrt{3}}{2}\cos(\theta) - \frac{1}{2}\sin(\theta) \]
\[ \sin\left(\frac{\pi}{3} + \theta\right) = \sin\left(\frac{\pi}{3}\right)\cos(\theta) + \cos\left(\frac{\pi}{3}\right)\sin(\theta) = \frac{\sqrt{3}}{2}\cos(\theta) + \frac{1}{2}\sin(\theta) \]

Now substituting these into the left-hand side:
\[ \sin(\theta) \left(\frac{\sqrt{3}}{2}\cos(\theta) - \frac{1}{2}\sin(\theta)\right) \left(\frac{\sqrt{3}}{2}\cos(\theta) + \frac{1}{2}\sin(\theta)\right) \]

This simplifies to:
\[ \sin(\theta) \left(\frac{3}{4}\cos^2(\theta) - \frac{1}{4}\sin^2(\theta)\right) \]

Using the identity \(\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)\), we can confirm that:
\[ \frac{1}{4} \sin(3\theta) = \frac{1}{4}(3\sin(\theta) - 4\sin^3(\theta)) \]

Thus, we have shown that Statement 2 is correct.

### Step 2: Analyzing Statement 1

Now we analyze Statement 1:
\[ \sin\left(\theta - \frac{\pi}{4}\right) \sin\left(\theta - \frac{7\pi}{12}\right) \sin\left(\theta + \frac{\pi}{12}\right) \]
We need to show that this expression lies between \(-\frac{1}{4\sqrt{2}}\) and \(\frac{1}{4}\).

Using the same substitution as in Statement 2, we can express this in terms of \(\sin(3\theta)\):
\[ \sin\left(\theta - \frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}(\sin(\theta) - \cos(\theta)) \]
\[ \sin\left(\theta - \frac{7\pi}{12}\right) = \sin\left(\frac{\pi}{3} - \theta\right) \]
\[ \sin\left(\theta + \frac{\pi}{12}\right) = \sin\left(\frac{\pi}{3} + \theta\right) \]

Thus, we can use the results from Statement 2 to show that:
\[ \sin\left(\theta - \frac{\pi}{4}\right) \sin\left(\theta - \frac{7\pi}{12}\right) \sin\left(\theta + \frac{\pi}{12}\right) = \frac{1}{4} \sin(3\theta - \frac{3\pi}{4}) \]

### Step 3: Finding the Range

Now we need to find the range of \(\sin(3\theta - \frac{3\pi}{4})\):
- The maximum value of \(\sin(x)\) is 1 and the minimum is -1.
- Thus, the maximum of \(\frac{1}{4} \sin(3\theta - \frac{3\pi}{4})\) is \(\frac{1}{4}\) and the minimum is \(-\frac{1}{4}\).

To find the specific bounds:
\[ -\frac{1}{4\sqrt{2}} < \frac{1}{4} \sin(3\theta - \frac{3\pi}{4}) < \frac{1}{4} \]

### Conclusion

Both statements are correct. Therefore, the final conclusion is that both Statement 1 and Statement 2 are true.