Given: #" "# #Sin 10*Sin 30*Sin*50*sin 70#
Put #\ \ \ # #Sin30=1/2#
#=\ (1/ 2) Sin 10 * Sin 50 * sin 70#
Multiply and divide with #2Cos 10#
#=\ \frac{2\ Cos10\cdot Sin10\cdot Sin\ 50\cdot sin\ 70}{2\cdot 2Cos10}#
Apply the double angle formula #\ \ ##2Sin\ x\ Cos\ x\ =\ Sin\ 2x\ #
#=\ \frac{Sin(2*10)\cdot Sin\ 50\cdot sin\ 70}{4Cos10}#
#=\ \frac{Sin\ 20\cdot Sin\ 50\cdot sin\ 70}{4Cos10}#
Rewrite #\ \ \ # #sin(70)=sin(90-20)=cos(20)# #\ \ \ # because #\ \ \ ##sin(\pi/2-x)=cos(x)#
#=\ \frac{Sin\ 20\cdot Cos\ 20\cdot sin\ 50}{4Cos10}#
Multiply and divide by 2 and apply the double angle formula again:
#=\ \frac{2\cdot Sin\ 20\cdot Cos\ 20\cdot sin\ 50}{2\cdot 4Cos10}#
#=\ \frac{Sin\ 40\cdot sin\ 50}{8Cos10}#
Repeat the same steps, #\ \ \ # #sin(50)=sin(90-40)=cos(40)#
#=\ \frac{2\cdot Sin\ 40\cdot cos\ 40}{2\cdot 8Cos10}#
#=\ \frac{Sin\ 80}{16Cos10}#
Apply #\ \ \ ##cos(10)=cos(90-80)=sin(80)# #\ \ \ # because #\ \ \ ##cos(\pi/2-x)=sin(x)#
#=\ \frac{Sin\ 80}{16sin\ 80}#
Cancel out #sin\ 80#
#=1/16#
That's it!
