Satement-1 : If `secx + cos x =2` then `(sec x)^(n)+(cos x^(n)=2`.
because
Statement-2 : If `A+B=2` and `AB=1` then `A=B=1`.

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.

Answer

To solve the problem, we need to analyze the two statements given and determine their validity.

### Step 1: Analyze Statement 2
We are given that if \( A + B = 2 \) and \( AB = 1 \), then we need to show that \( A = B = 1 \).

1. Start with the equations:
\[ A + B = 2 \]
\[ AB = 1 \]

2. From the first equation, we can express \( B \) in terms of \( A \):
\[ B = 2 - A \]

3. Substitute \( B \) in the second equation:
\[ A(2 - A) = 1 \]
\[ 2A - A^2 = 1 \]

4. Rearranging gives us a quadratic equation:
\[ A^2 - 2A + 1 = 0 \]

5. This can be factored as:
\[ (A - 1)^2 = 0 \]

6. Thus, we find:
\[ A - 1 = 0 \implies A = 1 \]

7. Substituting back to find \( B \):
\[ B = 2 - A = 2 - 1 = 1 \]

So, we have shown that \( A = B = 1 \). Therefore, Statement 2 is true.

### Step 2: Analyze Statement 1
Now we need to show that if \( \sec x + \cos x = 2 \), then \( \sec^n x + \cos^n x = 2 \).

1. Start with the equation:
\[ \sec x + \cos x = 2 \]

2. Recall that \( \sec x = \frac{1}{\cos x} \). Substitute this into the equation:
\[ \frac{1}{\cos x} + \cos x = 2 \]

3. Multiply through by \( \cos x \) (assuming \( \cos x \neq 0 \)):
\[ 1 + \cos^2 x = 2 \cos x \]

4. Rearranging gives us:
\[ \cos^2 x - 2 \cos x + 1 = 0 \]

5. This can be factored as:
\[ (\cos x - 1)^2 = 0 \]

6. Thus, we find:
\[ \cos x - 1 = 0 \implies \cos x = 1 \]

7. Since \( \sec x = \frac{1}{\cos x} \), we have:
\[ \sec x = \frac{1}{1} = 1 \]

8. Now we can evaluate \( \sec^n x + \cos^n x \):
\[ \sec^n x + \cos^n x = 1^n + 1^n = 1 + 1 = 2 \]

Thus, we have shown that \( \sec^n x + \cos^n x = 2 \). Therefore, Statement 1 is also true.

### Conclusion
Both statements are true, and Statement 2 serves as a correct explanation for Statement 1.