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cos^4(pi/8)+cos^4((3pi)/8)+cos^4((5pi)/8...

`cos^4(pi/8)+cos^4((3pi)/8)+cos^4((5pi)/8)+cos^4((7pi)/8)=`

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Given expression, `cos^(4)""(pi)/(8)+cos^(4)""(3pi)/(8)+cos^(4)""(5pi)/(8)+cos^(4)""(7pi)/(8)`
`" "=cos^(4)""(pi)/(8)+cos^(4)""(3pi)/(8)+cos^(4)(pi-(3pi)/(8))+cos^(4)(pi-(pi)/(8))`
`" "=cos^(4)""(pi)/(8)+cos^(4)""(3pi)/(8)+cos^(4)""(3pi)/(8)+cos^(4)""(pi)/(8)`
`" "=2[cos^(4)""(pi)/(8)+cos^(4)""(3pi)/(8)]=2[cos^(4)""(pi)/(8)+cos^(4)""((pi)/(2)-(pi)/(8))]`
`" "=2[cos^(4)""(pi)/(8)+sin^(4)""(pi)/(8)]`
`" "=2[(cos^(2)""(pi)/(8)+sin^(2)""(pi)/(8))^(2)-2cos^(2)""(pi)/(8)*sin^(2)""(pi)/(8)]`
`" "=2[1-2cos^(2)""(pi)/(8)*sin^(2)""(pi)/(8)] = 2-(2sin""(pi)/(8)*cos"(pi)/(8))^(2)`
` " "=2-(sin""(2pi)/(8))^(2)=2-((1)/(sqrt(2)))^(2)`
`" "=2-(1)/(2)=(3)/(2)`
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