Answer to Question #13549 in Trigonometry for Basu

Question

Show that cos20°cos40°cos60°cos80°=1/16

Answer 1

cos20*cos40*cos60*cos80=
At first multiply the expression by sin20 and divide by sin20:

= sin20.cos20.cos40.cos60.cos80/sin20



Since sin20*cos20=1/2 (sin 0+sin40) we obtain

=sin40*cos40*cos60*cos80/{2 sin20}

Also sin40*cos40=1/2 sin80

so we get
=sin80*cos80*cos60/{4 sin20}



Again sin80*cos80=1/2 sin160

and finally
=sin160*cos60/{8sin20}=1/16