To prove that sin(2x/3) + cos^2(2x/3) = 1, we can first rewrite cos^2(2x/3) in terms of sin(2x/3) using the Pythagorean identity:

cos^2(2x/3) = 1 - sin^2(2x/3)

Then, substitute this expression back into the original equation:

sin(2x/3) + (1 - sin^2(2x/3)) = 1

Now we have:

sin(2x/3) + 1 - sin^2(2x/3) = 1

Rearranging terms, we get:

1 - sin^2(2x/3) + sin(2x/3) = 1

Since sin^2(2x/3) + sin(2x/3) = 1 (from the Pythagorean trigonometric identity), we have:

1 = 1

Therefore, we have proved that sin(2x/3) + cos^2(2x/3) = 1.