Question #13549

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

Expert's answer

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

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

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