Statement I The curve `y=81^(sin^(2)x)+81^(cos^(2)x)-30` intersects X-axis at eight points in the region `-pi le x le pi`.
Statement II The curve `y=sinx` or `y=cos x` intersects the X-axis at infinitely many points.

Options

A. Both Statement I and Statement II are individually true and R is the correct explanation of Statement I.
B. Both Statement I and Statement II are individually true but Statement II is not the correct explanaton of Statement I.
C. Statement I is true but Statement II is false.
D. Statement I is false but Statement II is true.

Answer

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

### Statement I:
The curve \( y = 81^{\sin^2 x} + 81^{\cos^2 x} - 30 \) intersects the X-axis at eight points in the region \( -\pi \leq x \leq \pi \).

1. **Understanding the Function**:
The function can be rewritten using the identity \( \sin^2 x + \cos^2 x = 1 \):
\[ y = 81^{\sin^2 x} + 81^{\cos^2 x} - 30 \]
Since \( \sin^2 x + \cos^2 x = 1 \), we can express \( 81^{\cos^2 x} \) as \( 81^{1 - \sin^2 x} \).

2. **Finding Critical Points**:
To find the points where the curve intersects the X-axis, we set \( y = 0 \):
\[ 81^{\sin^2 x} + 81^{\cos^2 x} = 30 \]
This can be simplified to:
\[ 81^{\sin^2 x} + 81^{1 - \sin^2 x} = 30 \]

3. **Using Substitution**:
Let \( a = 81^{\sin^2 x} \). Then, \( 81^{\cos^2 x} = \frac{81}{a} \). The equation becomes:
\[ a + \frac{81}{a} = 30 \]
Multiplying through by \( a \) gives:
\[ a^2 - 30a + 81 = 0 \]

4. **Solving the Quadratic**:
We can use the quadratic formula to find \( a \):
\[ a = \frac{30 \pm \sqrt{30^2 - 4 \cdot 1 \cdot 81}}{2 \cdot 1} \]
\[ = \frac{30 \pm \sqrt{900 - 324}}{2} \]
\[ = \frac{30 \pm \sqrt{576}}{2} \]
\[ = \frac{30 \pm 24}{2} \]
This results in two solutions:
\[ a_1 = 27 \quad \text{and} \quad a_2 = 3 \]

5. **Finding \( x \) Values**:
Now we need to find the values of \( x \) for which \( 81^{\sin^2 x} = 27 \) and \( 81^{\sin^2 x} = 3 \):
- For \( 81^{\sin^2 x} = 27 \):
\[ \sin^2 x = \frac{3}{4} \implies \sin x = \pm \frac{\sqrt{3}}{2} \]
This gives \( x = \frac{\pi}{3}, -\frac{\pi}{3}, \frac{2\pi}{3}, -\frac{2\pi}{3} \).
- For \( 81^{\sin^2 x} = 3 \):
\[ \sin^2 x = \frac{1}{4} \implies \sin x = \pm \frac{1}{2} \]
This gives \( x = \frac{\pi}{6}, -\frac{\pi}{6}, \frac{5\pi}{6}, -\frac{5\pi}{6} \).

6. **Counting the Points**:
The total points of intersection are:
- From \( 81^{\sin^2 x} = 27 \): 4 points
- From \( 81^{\sin^2 x} = 3 \): 4 points
- Total: \( 4 + 4 = 8 \)

Thus, Statement I is **true**.

### Statement II:
The curves \( y = \sin x \) and \( y = \cos x \) intersect the X-axis at infinitely many points.

1. **Analyzing the Functions**:
- The function \( y = \sin x \) intersects the X-axis at \( x = n\pi \) for \( n \in \mathbb{Z} \).
- The function \( y = \cos x \) intersects the X-axis at \( x = \frac{\pi}{2} + n\pi \) for \( n \in \mathbb{Z} \).

2. **Conclusion**:
Both functions intersect the X-axis infinitely many times. Thus, Statement II is also **true**.

### Final Conclusion:
Both statements are true.