Question

prove that cos20 cos 40 cos 60 cos 80=1/16.

Answer 1

Hey mate!!!

since cos 60 = 1/2

LHS=1/2 cos20cos40cos80

multiplying and dividing by 2,

=>1/4 cos80(2cos40cos20)

=1/4 cos80(cos(40+20) + cos(40-20))

= 1/4 cos80(cos60+cos20)

=1/4 cos80(1/2 + cos20)

opening the bracket:

=> 1/8 cos80 + 1/4 cos80cos20

multiplying and dividing [1/4 cos80cos20] by 2,

=> 1/8 cos80 + 1/8 (2cos80cos20)

=1/8 cos80 + 1/8 (cos100 + cos60)

=1/8 cos80 + 1/8 (cos100 + 1/2)

=1/8 cos80 + 1/8 cos100 + 1/16

 since cos100 = cos (180-80) = -cos80,

=> 1/8 cos80 + 1/8 (-cos80) +1/16

= 1/8 cos80 - 1/8 cos80 + 1/16

= 1/16 = RHS

HOPE IT HELPS.

Answer 2

hello mate

here is your answer

question--prove that cos20 cos 40 cos 60 cos 80=1/16.

solution--

since cos 60 = 1/2
LHS=1/2 cos20cos40cos80

multiplying and dividing by 2,

=>1/4 cos80(2cos40cos20)
=1/4 cos80(cos(40+20) + cos(40-20))
= 1/4 cos80(cos60+cos20)
=1/4 cos80(1/2 + cos20)
=> 1/8 cos80 + 1/4 cos80cos20

multiplying and dividing [1/4 cos80cos20] by 2,

=> 1/8 cos80 + 1/8 (2cos80cos20)
=1/8 cos80 + 1/8 (cos100 + cos60)
=1/8 cos80 + 1/8 (cos100 + 1/2)
=1/8 cos80 + 1/8 cos100 + 1/16

 since cos100 = cos (180-80) = -cos80,
=> 1/8 cos80 + 1/8 (-cos80) +1/16
= 1/8 cos80 - 1/8 cos80 + 1/16
= 1/16 = RHS

I hope helps you