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Question
The value of `cos pi/5 cos (2pi)/5 cos (4pi)/5 cos (8pi)/5` is ______.
Options
`1/16`
0
`(-1)/8`
`(-1)/16`
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Solution
The value of `cos pi/5 cos (2pi)/5 cos (4pi)/5 cos (8pi)/5` is `(-1)/16`.
Explanation:
We have `cos pi/5 cos (2pi)/5 cos (4pi)/5 cos (8pi)/5`
= `1/(2sin pi/5) 2sin pi/5 cos pi/5 cos (2pi)/5 cos (4pi)/5 cos (8pi)/5`
= `1/(2sin pi/5) sin (2pi)/5 cos (2pi)/5 cos (4pi)/5 cos (8pi)/5`
= `1/(4sin pi/5) sin (4pi)/5 cos (4pi)/5 cos (8pi)/5`
= `1/(8sin pi/5) sin (8pi)/5 cos (8pi)/5`
= `(sin (16pi)/5)/(16sin pi/5)`
= `(sin(3pi + pi/5))/(16sin pi/5)`
= `(-sin pi/5)/(16sin pi/5)`
= `(-1)/16`
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