To solve the problem, we need to analyze the two statements given and determine their validity based on the properties of inverse trigonometric functions.
### Step 1: Analyze Statement I
The first statement is:
\[ y = \tan^{-1}(\tan x) \text{ and } y = \cos^{-1}(\cos x) \text{ does not have any solution if } x \in \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \]
1. **Understanding \( y = \tan^{-1}(\tan x) \)**:
- The function \( \tan^{-1}(\tan x) \) returns \( x \) when \( x \) is in the interval \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).
- For \( x \in \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \), \( \tan x \) is negative, and thus \( \tan^{-1}(\tan x) \) will yield a value in the range \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), specifically \( x - \pi \).
2. **Understanding \( y = \cos^{-1}(\cos x) \)**:
- The function \( \cos^{-1}(\cos x) \) returns \( x \) when \( x \) is in the interval \( [0, \pi] \).
- For \( x \in \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \), \( \cos x \) is negative, and thus \( \cos^{-1}(\cos x) \) will yield \( 2\pi - x \) for \( x \in [\pi, \frac{3\pi}{2}] \) and \( x \) for \( x \in \left(\frac{\pi}{2}, \pi\right) \).
### Step 2: Determine if there is a solution
- For \( x \in \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \):
- From the analysis, we have:
- \( y = \tan^{-1}(\tan x) = x - \pi \)
- \( y = \cos^{-1}(\cos x) = 2\pi - x \) for \( x \in [\pi, \frac{3\pi}{2}] \) and \( y = x \) for \( x \in \left(\frac{\pi}{2}, \pi\right) \).
- The ranges of \( y \):
- \( y = x - \pi \) gives values in \( \left(-\frac{\pi}{2} - \pi, \frac{3\pi}{2} - \pi\right) = \left(-\frac{3\pi}{2}, \frac{\pi}{2}\right) \).
- \( y = 2\pi - x \) gives values in \( [\pi, \frac{3\pi}{2}] \) which translates to \( [\pi, \frac{3\pi}{2}] \).
### Step 3: Check for intersections
- The ranges \( \left(-\frac{3\pi}{2}, \frac{\pi}{2}\right) \) and \( [\pi, \frac{3\pi}{2}] \) do not intersect.
- Thus, there is no solution for the equations in the specified interval.
### Conclusion for Statement I
- Therefore, Statement I is **true**.
### Step 4: Analyze Statement II
The second statement is:
\[ y = \tan^{-1}(\tan x) = x - \pi, \text{ for } x \in \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \]
and
\[ y = \cos^{-1}(\cos x) = \begin{cases}
2\pi - x, & x \in [\pi, \frac{3\pi}{2}] \\
x, & x \in \left(\frac{\pi}{2}, \pi\right)
\end{cases} \]
- As analyzed previously, both functions yield values that do not intersect in the specified ranges.
### Conclusion for Statement II
- Therefore, Statement II is also **true**.
### Final Answer
Both Statement I and Statement II are true.
