Question

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

Answer 1

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

Answer 2

ANSWER

Given=>

cos20 cos 40 cos 60 cos 80=$\dfrac{1}{16}$

Now.

In LHS

cos20 cos 40 cos 60 cos 80

=>(cos20°× cos40°)cos60°×cos80°

=>$\dfrac{1}{2}$[cos(20° + 40°)

+ cos(20° – 40°)]×$\dfrac{1}{2}$ ×cos80°

=>$\dfrac{1}{4}$[cos60° +cos (-20°)] cos80°

=>$\dfrac{1}{4}$ [cos60°cos80° + cos20°×cos80°]

=>{1/4}×[{1/2}×cos80°+

1/2×{cos(20°+80°)+cos(20°-80°)]

=>$\dfrac{1}{8}$[cos80° + {cos100° + cos(-60°)}]

=>$\dfrac{1}{8}$[cos80° + cos100° + cos60°]

=>$\dfrac{1}{8}$[cos80° +cos(180° – 80°) +cos60°]

=>$\dfrac{1}{8}$[cos80° – cos80° + cos60°]

=>$\dfrac{1}{8}$ ×cos60°

=>$\dfrac{1}{8}$ × $\dfrac{1}{2}$

=>$\dfrac{1}{16}$.

Hence,

LHS =RHS.

HENCE PROVED