<Polygon-2024-268>, hint

Let <CAD=x, hten <ADC=2x, <ACD=180-3x. 1/sinx=AC/sin2x=AD/sin3x

1/AC+1/AD =sinx(1/sin2x+1/sin3x). since 7x=180, 3x=180-4x, sin3x=sin4x

sinx(1/sin2x+1/sin4x), sin4x=2sin2x cos2x.  cos2x=2cos^2x-1

sin3x=sin4x  sin3x=3sinx-4sin^3x ...> 4cos^2x-1=8cos^3x-4cosx...<1>

sinx(1/sin2x+1/sin4x)=(4cos^2x-1)/(8cos^3x-4cosx)=1