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Prove that (4cos(2pi)/7.cospi/7-1=2cos(2...

Prove that `(4cos(2pi)/7.cospi/7-1=2cos(2pi)/7)`.

Text Solution

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LHS `=4 cos ""(2pi)/(7).cos"" (pi)/(7)-1`
`=(sin""(pi)/(7).4 cos ""(pi)/(7).cos((2pi)/(7)))/(sin""(pi)/(7))-1`
`=(2sin""(2pi)/(7)cos ""(2pi)/(7))/(sin""(pi)/(7))-1`
`=(sin ""(4pi)/(7))/(sin""(pi)/(7))-1`
`=(3sin""(pi)/(7)-4sin^(3)""(pi)/(7))/(sin""(pi)/(7))-1`
`3-4sin^(2)""(pi)/(7)-1`
`=2[1-2sin^(2)""(pi)/(7)]`
`=2cos""(2pi)/(7)=RHS`.
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