//part of the usings
using System.Runtime.InteropServices;
 
//declares
[DllImport("user32.dll")]
private static extern bool PostMessage(
IntPtr hWnd,// handle to destination window
Int32 msg,// message
Int32 wParam,// first message parameter
Int32 lParam // second message parameter
);
 
const Int32 WM_LBUTTONDOWN = 0x0201;
 
//method to call dropdown
private void button1_Click(object sender,EventArgs e)
{
    Int32 x = dateTimePicker1.Width - 10;
    Int32 y = dateTimePicker1.Height / 2;
    Int32 lParam = x + y * 0x00010000;
 
    PostMessage(dateTimePicker1.Handle,WM_LBUTTONDOWN,1,lParam);
 
}