A

Step 1: The given equation of the stationary wave is represented by y = a \, ext{sin} \, ext{ωt} \, ext{cos} \, kx.
Step 2: This equation can be derived from the superposition of two progressive waves traveling in opposite directions. The progressive waves can be expressed as:
y_1 = A \, ext{sin} \, ( ext{ωt} + kx) and y_2 = A \, ext{sin} \, ( ext{ωt} - kx).
Step 3: By using the trigonometric identity for sine addition, we can combine these two equations:
y = y_1 + y_2 = A \, [\text{sin} \, ( ext{ωt} + kx) + \text{sin} \, ( ext{ωt} - kx)].
Step 4: This reduces to y = 2A \, ext{sin} \, ext{ωt} \, ext{cos} \, kx when applying the sum-to-product identity. This indicates that the amplitude of the stationary wave is 2A.
Step 5: The amplitude of each progressive wave, therefore, is half of this stationary wave amplitude: A = \frac{2A}{2} = A.
Step 6: Hence, for y = a \, ext{sin} \, ext{ωt} \, ext{cos} \, kx, the amplitude of the component progressive wave is indeed \frac{a}{2}.
Therefore, the answer is A.