Statement-1: `lim_(xrarr oo) (cos (pi)/(x))^x=1`
Statement-2: `lim_(xrarr oo) -pi tan. (pi)/(x)=0`

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 given problem, we will analyze both statements step by step.

### Statement 1:
We need to evaluate the limit:
\[ L = \lim_{x \to \infty} \left( \cos\left(\frac{\pi}{x}\right) \right)^x \]

**Step 1: Take the natural logarithm of both sides.**
\[ \log L = \lim_{x \to \infty} x \log\left(\cos\left(\frac{\pi}{x}\right)\right) \]

**Hint:** Taking the logarithm simplifies the exponentiation in limits.

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**Step 2: Rewrite the limit.**
\[ \log L = \lim_{x \to \infty} \frac{\log\left(\cos\left(\frac{\pi}{x}\right)\right)}{\frac{1}{x}} \]

**Hint:** This form allows us to apply L'Hôpital's Rule since it is in the form \( \frac{0}{0} \).

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**Step 3: Apply L'Hôpital's Rule.**
Differentiate the numerator and denominator:
- The derivative of the numerator \( \log\left(\cos\left(\frac{\pi}{x}\right)\right) \) is:
\[ \frac{d}{dx} \log\left(\cos\left(\frac{\pi}{x}\right)\right) = -\tan\left(\frac{\pi}{x}\right) \cdot \left(-\frac{\pi}{x^2}\right) = \frac{\pi \tan\left(\frac{\pi}{x}\right)}{x^2} \]
- The derivative of the denominator \( \frac{1}{x} \) is:
\[ -\frac{1}{x^2} \]

Now we can rewrite the limit:
\[ \log L = \lim_{x \to \infty} \frac{\frac{\pi \tan\left(\frac{\pi}{x}\right)}{x^2}}{-\frac{1}{x^2}} = \lim_{x \to \infty} -\pi \tan\left(\frac{\pi}{x}\right) \]

**Hint:** L'Hôpital's Rule is useful for resolving indeterminate forms.

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**Step 4: Evaluate the limit.**
As \( x \to \infty \), \( \frac{\pi}{x} \to 0 \), and thus \( \tan\left(\frac{\pi}{x}\right) \to 0 \):
\[ \log L = \lim_{x \to \infty} -\pi \tan\left(\frac{\pi}{x}\right) = 0 \]

**Hint:** Knowing the behavior of trigonometric functions as their argument approaches zero is crucial.

---

**Step 5: Exponentiate to find \( L \).**
\[ L = e^0 = 1 \]

Thus, Statement 1 is true:
\[ \lim_{x \to \infty} \left( \cos\left(\frac{\pi}{x}\right) \right)^x = 1 \]

---

### Statement 2:
We need to evaluate the limit:
\[ \lim_{x \to \infty} -\pi \tan\left(\frac{\pi}{x}\right) \]

**Step 6: Evaluate the limit directly.**
As \( x \to \infty \), \( \frac{\pi}{x} \to 0 \), and thus \( \tan\left(\frac{\pi}{x}\right) \to 0 \):
\[ \lim_{x \to \infty} -\pi \tan\left(\frac{\pi}{x}\right) = -\pi \cdot 0 = 0 \]

Thus, Statement 2 is also true:
\[ \lim_{x \to \infty} -\pi \tan\left(\frac{\pi}{x}\right) = 0 \]

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### Conclusion:
Both statements are true. However, Statement 2 is not a correct explanation for Statement 1. Therefore, the correct option is that Statement 1 is true, Statement 2 is true, but Statement 2 is not a correct explanation for Statement 1.

**Final Answer:**
- Statement 1 is true.
- Statement 2 is true.
- Statement 2 is not a correct explanation for Statement 1.