Question

The greatest value of sinA.cosA is

Answer 1

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

Answer 2

Remark
I can't imagine you having to solve this question any other way than by graphing it, if you are in middle school mathematics.

Step One
Go to Desmos. Input y = sin(A)*Cos(A) on the input bar.

Step Two
Check to see where the highest point is. It should be somewhere around 45 degrees or pi/4 which is 0.7854 on the x axis.

Step Three
On my graph, you can click on the greatest point on the red line. It should come back with (pi/4,0.5) That is the highest point between 0 and 90 degrees. Depending on what your graph looks like, there is another such point a (5*pi/4,0.5)

Note
The other responder did this question using Trigonometry. If you understand the answer, I would follow it. If not then this is another way to do it.