help

sin (10°) sin (50°) sin(70°)  =

sin(70°)sin(50°)sin(10°)  =         [ because   sin A   = cos (90 - A)  we can write  ]

cos(20°) cos(40°)cos(80°)

Note that

sin (2A)  =  2 sinA cos A       solve for   cos A     and we have that

cos A =   sin 2A

              _______

               2 sin A

So

cos (20°)   =    sin (40°)

                       _________

                         2 sin (20°)

cos(40°)  =   sin (80°)

                   _________

                      2sin(40°)

cos(80°)  =   sin (160°)

                    __________

                     2sin(80°)

Multiplying these together, we have that

sin (40°)    *     sin ( 80°)      *    sin (160°)

_______        _________        __________  =  

2sin(20°)         2 sin(40°)           2sin(80°)

sin (160°)

_________   =                     [  note  that sin 20°  =  sin (180 - 20°)  = sin 160°]

8 sin(20°)

1

__

 8

BTW......this  is  the application of something known as " Morrie's Law"

The name is due to the famous physicist Richard Feynman, who used to refer to the identity under that name. Feynman picked that name because he learned it during his childhood from a boy with the name Morrie Jacobs and afterwards remembered it for all of his life.