1/sin40°+1/tan80°
=(2*cos40°/sin80°)+(cos80°/sin80°)
=(2*cos40°+cos80°)/sin80°
=[(cos80°+cos40°)+cos40°]/cos10°
=(cos40°+cos20°)/cos10°
=[(√3)cos10°]/cos10°
=√3
1/sin40°+1/tan80°
=(2*cos40°/sin80°)+(cos80°/sin80°)
=(2*cos40°+cos80°)/sin80°
=[(cos80°+cos40°)+cos40°]/cos10°
=(cos40°+cos20°)/cos10°
=[(√3)cos10°]/cos10°
=√3