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!
