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Question
Prove the following:
`cos (2pi)/15 cos (4pi)/15cos (8pi)/15cos (16pi)/15 = 1/16`
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Solution
L.H.S. = `cos (2pi)/15 cos (4pi)/15cos (8pi)/15cos (16pi)/15`
= `((2sin (2pi)/15*cos (2pi)/15)*cos (4pi)/15*cos (8pi)/15*cos (16pi)/15)/(2sin (2pi)/15)`
= `(sin (4pi)/15*cos (4pi)/15*cos (8pi)/15*cos (16pi)/15)/(2sin (2pi)/15)`
= `((2sin (4pi)/15*cos (4pi)/15)cos (8pi)/15*cos (16pi)/15)/(4sin (2pi)/15)`
= `(sin (8pi)/15*cos (8pi)/15*cos (16pi)/15)/(4sin (2pi)/15)`
= `((2sin (8pi)/15*cos (8pi)/15)cos (16pi)/15)/(8sin (2pi)/15)`
= `(sin (16pi)/15*cos (16pi)/15)/(8sin (2pi)/15)`
= `(2sin (16pi)/15*cos (16pi)/15)/(16sin (2pi)/15)`
= `(sin (32pi)/15)/(16sin (2pi)/15)`
= `(sin(2pi + (2pi)/15))/(16sin (2pi)/15)`
= `(sin (2pi)/15)/(16sin (2pi)/15)`
= `1/16`
= R.H.S.
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