To solve this trigonometric equation, we can combine like terms and use trigonometric identities to simplify it.

First, let's rewrite the equation (√2)sin(3π/2 - x) * sin(x) = cos(x) as:

(√2)sin(3π/2 - x) * sin(x) = cos(x)

Using the trigonometric identity sin(π/2 - θ) = cos(θ), we can rewrite sin(3π/2 - x) as cos(x).

(√2)cos(x) * sin(x) = cos(x)

Now, let's simplify the equation:

(√2)cos(x) * sin(x) = cos(x)

√2 cos(x) sin(x) = cos(x)

Now, we can use the identity sin(2θ) = 2sin(θ)cos(θ) to simplify further:

√2 * sin(2x) = cos(x)

Finally, we can divide both sides by √2 to get:

sin(2x) = cos(x) / √2

Since sin(2x) = 2sin(x)cos(x), we can rewrite the equation as:

2sin(x)cos(x) = cos(x) / √2

Next, we can divide both sides by cos(x) to isolate sin(x):

2sin(x) = 1 / √2

Finally, solving for sin(x):

sin(x) = 1 / √2 / 2

sin(x) = 1 / 2√2

Therefore, the solution to the trigonometric equation (√2)sin(3π/2 - x) * sin(x) = cos(x) is sin(x) = 1 / 2√2.