Statement-1 : If ` e^(itheta)=cos theta +isintheta` and the value of ` e^(iA).e^(iB).e^(iC)` is equal to -1, where A,B,C are the angles of a triangle.
and
Statement -2 : In any ` triangleABC , A + B +C = 180^(@)`

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 Flase, Statement -2 is True

Answer

To solve the problem, we need to analyze the two statements provided and establish their validity step by step.

### Step 1: Understanding the first statement
The first statement is:
\[ e^{iA} \cdot e^{iB} \cdot e^{iC} = -1 \]
where \( A, B, C \) are the angles of a triangle.

Using the property of exponents, we can combine the left side:
\[ e^{iA} \cdot e^{iB} \cdot e^{iC} = e^{i(A + B + C)} \]

### Step 2: Relating angles of a triangle
From geometry, we know that the sum of the angles in a triangle is:
\[ A + B + C = 180^\circ \]

### Step 3: Substituting the sum of angles
Substituting this into our expression gives:
\[ e^{i(A + B + C)} = e^{i \cdot 180^\circ} \]

### Step 4: Evaluating \( e^{i \cdot 180^\circ} \)
Using Euler's formula:
\[ e^{i \theta} = \cos \theta + i \sin \theta \]
we can evaluate:
\[ e^{i \cdot 180^\circ} = \cos(180^\circ) + i \sin(180^\circ) \]

### Step 5: Finding the values of cosine and sine
We know:
\[ \cos(180^\circ) = -1 \]
\[ \sin(180^\circ) = 0 \]

Thus:
\[ e^{i \cdot 180^\circ} = -1 + 0i = -1 \]

### Step 6: Conclusion for Statement 1
Since we have shown that:
\[ e^{iA} \cdot e^{iB} \cdot e^{iC} = -1 \]
is true when \( A + B + C = 180^\circ \), the first statement is correct.

### Step 7: Understanding the second statement
The second statement is:
"In any triangle, \( A + B + C = 180^\circ \)."

This is a fundamental property of triangles and is always true.

### Final Conclusion
Both statements are correct, and Statement 2 provides a valid explanation for Statement 1.