The greatest value of sinA. cosA is

Use the trig identity

2*sin (A) * cos (A) = sin (2*A)

to get

sin (A) * cos (A) = (1/2) * sin (2*A)

So to find the max of sin (A) * cos (A), we can find the max of (1/2) * sin (2*A)

It turns out that sin (x) maxes out at 1 where x can be any expression you want. In this case, x = 2*A.

So (1/2) * sin (2*A) maxes out at (1/2) * 1 = 1/2 = 0.5

The greatest value of sin (A) * cos (A) is 1/2 = 0.5