FIND

x lies in third quadrant means pi ≤ x ≤3*pi/2

sinx = -4/5  (since sin is negative in this range)

then pi/2≤ x/2 ≤3*pi/4   and    2*pi≤ 2x ≤3*pi

we know that |cos x| = 2cos² x/2 -1  =>  cos(x/2) = 2/5

but since pi/2≤ x/2 ≤3*pi/4 in that range cos is negative cos(x/2) = -2/5

similarly sin is positive in both the range sin(x/2) = 1/5

and sin2x = 2 sinx cos x = 2(-3/5)(-4/5) = 24/25