Assertion: Two waves `y_1 = A sin (omegat - kx)` and `y_2 = A cos(omegat-kx)` are superimposed, then `x=0` becomes a node.
Reason: At node net displacement due to waves should be zero.

Options

A. (a)Both Assertion and Reason are true and the Reason is correct explanation of the Assertion.
B. (b)Both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
C. (c)Assertion is true, but the Reason is false.
D. (d)Assertion is false but the Reason is true.

Explanation

To solve the problem, we need to analyze the assertion and reason given in the question regarding the superposition of two waves.

### Step-by-Step Solution:

1. **Identify the Waves**:
We have two waves given:
- \( y_1 = A \sin(\omega t - kx) \)
- \( y_2 = A \cos(\omega t - kx) \)

2. **Superimpose the Waves**:
The resultant wave \( y \) when these two waves are superimposed can be expressed as:
\[ y = y_1 + y_2 = A \sin(\omega t - kx) + A \cos(\omega t - kx) \]

3. **Evaluate at \( x = 0 \)**:
Substitute \( x = 0 \) into the equation:
\[ y = A \sin(\omega t) + A \cos(\omega t) \]
This simplifies to:
\[ y = A(\sin(\omega t) + \cos(\omega t)) \]

4. **Determine Conditions for a Node**:
A node occurs where the net displacement is zero. For \( y \) to be zero, we need:
\[ A(\sin(\omega t) + \cos(\omega t)) = 0 \]
This means:
\[ \sin(\omega t) + \cos(\omega t) = 0 \]

5. **Analyze the Condition**:
The equation \( \sin(\omega t) + \cos(\omega t) = 0 \) can be rewritten as:
\[ \sin(\omega t) = -\cos(\omega t) \]
This implies:
\[ \tan(\omega t) = -1 \]
This condition does not hold true for all values of \( t \). Therefore, there are specific instances where this can be true, but not universally at \( x = 0 \).

6. **Conclusion**:
Since the net displacement at \( x = 0 \) does not become zero for all time \( t \), \( x = 0 \) does not become a node. Thus, the assertion is incorrect. The reason provided is correct as it states that at a node, the net displacement should be zero.

### Final Answer:
- Assertion: False
- Reason: True