Assertion (A) : In `DeltaABC ` , right angled at C and `angle A = angle B` then cos A = cos B .
Reason (R ) : In a triangle , equal opposite sides have equal opposite angles .

Options

A. Both A and R are true and R is the correct explanation of A.
B. Both A and R are true and R is NOT the correct explanation of A.
C. A is true but R is false
D. A is false but R is true

Answer

To solve the assertion and reason question, we will analyze both the assertion (A) and the reason (R) step by step.

### Step-by-Step Solution:

1. **Understanding the Assertion (A)**:
- We have a triangle \( \Delta ABC \) which is right-angled at \( C \).
- It is given that \( \angle A = \angle B \).

2. **Using the Triangle Sum Property**:
- The sum of angles in any triangle is \( 180^\circ \).
- Since \( \angle C = 90^\circ \), we can express the sum of the other two angles as:
\[ \angle A + \angle B + \angle C = 180^\circ \]
\[ \angle A + \angle B + 90^\circ = 180^\circ \]
- Rearranging gives:
\[ \angle A + \angle B = 90^\circ \]

3. **Substituting the Equal Angles**:
- Since \( \angle A = \angle B \), we can substitute \( \angle B \) with \( \angle A \):
\[ \angle A + \angle A = 90^\circ \]
\[ 2\angle A = 90^\circ \]
- Dividing both sides by 2 gives:
\[ \angle A = 45^\circ \quad \text{and} \quad \angle B = 45^\circ \]

4. **Calculating Cosine Values**:
- Now we can find \( \cos A \) and \( \cos B \):
\[ \cos A = \cos 45^\circ = \frac{1}{\sqrt{2}} \]
\[ \cos B = \cos 45^\circ = \frac{1}{\sqrt{2}} \]
- Therefore, we conclude that:
\[ \cos A = \cos B \]

5. **Conclusion of Assertion**:
- The assertion \( \cos A = \cos B \) is true.

6. **Understanding the Reason (R)**:
- The reason states that in a triangle, equal opposite sides have equal opposite angles.
- This is a fundamental property of triangles known as the Isosceles Triangle Theorem.

7. **Connecting Reason to Assertion**:
- Since \( \angle A = \angle B \), the sides opposite to these angles are equal, confirming that the reason is also true.
- Therefore, the reason correctly explains the assertion.

### Final Conclusion:
- Both the assertion (A) and the reason (R) are true, and R is the correct explanation for A.