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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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LHS
`=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(pi-(3pi)/8)]^(4) + [cos(pi-pi/8)]^(4)`
`=cos^(4)pi/8+cos^(4)(3pi)/(8)+cos^(4)pi/8`
`=2[cos^(4)pi/8+cos^(4)(3pi)/(8)]`
`=1/2[(2cos^(2)pi/8)^(2)+(2cos^(2)(3pi)/8)^(2)]`
`=1/2[(1+cospi/4)^(2)+(1+cos(3pi)/4)^(2)]` `(therefore 2cos^(2)theta=1+cos2theta)`
`=1/2[(1+1/sqrt(2))^(2){1+cos(pi-pi/4)}^(2)]`
`=1/2[(1+1/sqrt(2))^(2)+(1-cospi/4)^(2)]`
`=1/2[(1+1/sqrt(2))^(2)+(1-1/sqrt(2))^(2)]`
`=1/2[1/2+2/sqrt(2)+2/sqrt(2)+1+1/sqrt(2)-2/sqrt(2)]`
`=3/2`= RHS Hence Proved.
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