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Prove that: cos^4pi/8+cos^4(3pi)/8+cos^4...

Prove that: `cos^4pi/8+cos^4(3pi)/8+cos^4(5pi)/8+cos^4(7pi)/8=3/2`

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`L.H.S. = 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)...[As cos(pi-theta) = costheta]`
`=2(cos^4((pi)/8)+cos^4((3pi)/8))`
`=2(cos^4((pi)/8)+sin^4(pi/2-(3pi)/8))`
`=2(cos^4((pi)/8)+sin^4((pi)/8))`
`=2((cos^2(pi/8)+sin^2(pi/8))^2 - 2sin^2(pi/8)cos^2(pi/8))`
`=2(1-(2sin(pi/8)cos(pi/8))^2/2)`
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  1. Prove that: cos^4pi/8+cos^4(3pi)/8+cos^4(5pi)/8+cos^4(7pi)/8=3/2

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