Statement (A) : If `tantheta+cottheta=2` , then for all `n epsilon N, tan^(n)theta+cot^(n)theta=2`
Reason (R ) : For all `n epsilon N, tan theta+cot theta=tan^(n)theta+cot^(n)theta`

Options

A. 1. Both A and R are correct and R is the correct explanation of A .
B. 2. Both A and R are correct but R is not a correct explanation of A .
C. 3. A is correct, R is incorrect .
D. 4. A is incorrect, R is correct .

Answer

To solve the problem, we need to analyze the given statement (A) and reason (R) regarding the trigonometric functions.

### Step 1: Analyze Statement (A)
The statement claims that if \( \tan \theta + \cot \theta = 2 \), then for all \( n \in \mathbb{N} \), \( \tan^n \theta + \cot^n \theta = 2 \).

#### Explanation:
- We know that \( \tan \theta + \cot \theta = 2 \) holds true when \( \theta = 45^\circ \) because:
\[ \tan 45^\circ = 1 \quad \text{and} \quad \cot 45^\circ = 1 \]
Thus, \( \tan 45^\circ + \cot 45^\circ = 1 + 1 = 2 \).

### Step 2: Check for \( n = 1 \)
For \( n = 1 \):
\[ \tan^1 \theta + \cot^1 \theta = \tan \theta + \cot \theta = 2 \]
This holds true.

### Step 3: Check for \( n = 2 \)
For \( n = 2 \):
\[ \tan^2 \theta + \cot^2 \theta = (\tan \theta + \cot \theta)^2 - 2 \tan \theta \cot \theta \]
Since \( \tan \theta \cot \theta = 1 \), we have:
\[ \tan^2 \theta + \cot^2 \theta = 2^2 - 2 = 4 - 2 = 2 \]
This also holds true.

### Step 4: General Case for \( n \)
For any natural number \( n \):
Using the identity:
\[ \tan^n \theta + \cot^n \theta = (\tan \theta + \cot \theta)(\tan^{n-1} \theta + \cot^{n-1} \theta) - \tan \theta \cot \theta (\tan^{n-2} \theta + \cot^{n-2} \theta) \]
This recursion shows that if \( \tan \theta + \cot \theta = 2 \), then \( \tan^n \theta + \cot^n \theta \) will also equal 2 for all \( n \in \mathbb{N} \).

### Conclusion for Statement (A)
Thus, Statement (A) is correct.

### Step 5: Analyze Reason (R)
The reason states that \( \tan \theta + \cot \theta = \tan^n \theta + \cot^n \theta \) for all \( n \in \mathbb{N} \).

#### Explanation:
- This is not true in general. The equality \( \tan \theta + \cot \theta = \tan^n \theta + \cot^n \theta \) does not hold for all angles \( \theta \) unless \( \theta = 45^\circ \).

### Conclusion for Reason (R)
Thus, Reason (R) is incorrect.

### Final Conclusion
- Statement (A) is true.
- Reason (R) is false.