Is it legal to carry a pistol while pheasant hunting provided any pheasants shot at and reduced to possession are taken with a shotgun ?

Yes, nothing in Idaho Code, Law or rule prohibits you from carrying a firearm. In fact Idaho Code 36-401 (g) states: Nothing contained herein shall be construed to prohibit citizens of the United States who are residents of the state of Idaho from carrying arms for the protection of life and property.