Let `theta in (pi//4,pi//2), ` then
Statement I ` (cos theta)^(sin theta ) lt ( cos theta )^(cos theta ) lt ( sin theta )^(cos theta )`
Statement II The equation ` e^(sin theta )-e^(-sin theta )=4` ha a unique solution.

Options

A. (a)Statement I is true , Statement II is true , Statement II is a correct explanation for Statement I.
B. (b)Statement I is true , Statement II is true , Statement II is not a correct explanation for Statement II.
C. (c)Statement I is true , Statement II is false
D. (d)Statement I is false , Statement II true .

Answer

To solve the problem, we need to analyze both statements provided in the question.

### Statement I:
We need to prove that \( (\cos \theta)^{\sin \theta} < (\cos \theta)^{\cos \theta} < (\sin \theta)^{\cos \theta} \) for \( \theta \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) \).

1. **Understanding the range of \( \theta \)**:
- In the interval \( \left( \frac{\pi}{4}, \frac{\pi}{2} \right) \):
- \( \cos \theta \) decreases from \( \frac{1}{\sqrt{2}} \) to \( 0 \).
- \( \sin \theta \) increases from \( \frac{1}{\sqrt{2}} \) to \( 1 \).

2. **Analyzing \( (\cos \theta)^{\sin \theta} \) and \( (\cos \theta)^{\cos \theta} \)**:
- Since \( \sin \theta > \cos \theta \) in this interval, we can compare the two expressions:
- \( (\cos \theta)^{\sin \theta} < (\cos \theta)^{\cos \theta} \) because the base \( \cos \theta \) is less than 1 and the exponent \( \sin \theta \) is greater than \( \cos \theta \).

3. **Analyzing \( (\cos \theta)^{\cos \theta} \) and \( (\sin \theta)^{\cos \theta} \)**:
- Since \( \sin \theta > \cos \theta \) in this interval, we can compare the two expressions:
- \( (\cos \theta)^{\cos \theta} < (\sin \theta)^{\cos \theta} \) because \( \sin \theta > \cos \theta \) and both bases are positive.

Thus, we conclude that:
\[ (\cos \theta)^{\sin \theta} < (\cos \theta)^{\cos \theta} < (\sin \theta)^{\cos \theta} \]
This means Statement I is true.

### Statement II:
We need to analyze the equation \( e^{\sin \theta} - e^{-\sin \theta} = 4 \).

1. **Rewriting the equation**:
- Let \( x = e^{\sin \theta} \). Then, the equation becomes:
\[ x - \frac{1}{x} = 4 \]
- Multiplying through by \( x \) gives:
\[ x^2 - 4x - 1 = 0 \]

2. **Finding the roots**:
- Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{4 \pm \sqrt{16 + 4}}{2} = \frac{4 \pm \sqrt{20}}{2} = 2 \pm \sqrt{5} \]

3. **Determining the nature of the solutions**:
- Since \( e^{\sin \theta} > 0 \), we discard the negative root \( 2 - \sqrt{5} \) as it is less than zero.
- The positive root \( 2 + \sqrt{5} \) is valid, and since \( e^{\sin \theta} \) is a strictly increasing function, this means there is a unique solution for \( \sin \theta \).

Thus, Statement II is also true.

### Conclusion:
Both statements are true, but Statement II does not provide a correct explanation for Statement I. Therefore, the answer is that Statement I is true and Statement II is true, but Statement II is not a correct explanation for Statement I.