化简cos^8(π/8)+cos^8(3π/8)+cos^8(5π/8)+cos^8(7π/8)

化简cos^8(π/8)+cos^8(3π/8)+cos^8(5π/8)+cos^8(7π/8)

cos^2(π/8)+sin^2(π/8)=1
2cosπ/8sinπ/8=sin(π/4)=√2/2
2cos^2(π/8)sin^2(π/8)=1/4
cos^4(π/8)+sin^4(π/8)
=(cos^2(π/8)+sin^2(π/8))^2-2cos^2(π/8)sin^2(π/8)
=1-1/4
=3/4
cos^8(π/8)+sin^8(π/8)
=(cos^4(π/8)+sin^4(π/8))^2-2cos^4(π/8)sin^4(π/8)
=9/16-1/32
=17/32
cos^8(π/8)+cos^8(3π/8)+cos^8(5π/8)+cos^8(7π/8)
=cos^8(π/8)+cos^8(π/2-π/8)+cos^8(π/2+π/8)+cos^8(π-π/8)
=cos^8(π/8)+sin^8(π/8)+sin^8(π/8)+cos^8(π/8)
=2[cos^8(π/8)+sin^8(π/8)]
=2*17/32=17/16

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