Prove that cos^4(π/8)+cos^4(3π/8)+cos^4(5π/8)+cos^4(7π/8)=3/2

Steps to prove cos^4(π/8)+cos^4(3π/8)+cos^4(5π/8)+cos^4(7π/8)=3/2

We know that 

cos^4(3π /8)=cos^4(π/2-π/8)=sin^4(π/8)

cos^4(5π/8)=cos^4(π/2+π/8)=sin^4(π/8)

cos^4(7π/8)=cos^4(π-π/8)=cos^4(π/8)

L.H.S

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

=cos^4(π/8)+sin^4(π/8)+sin^4(π/8)+cos^4(π/8)

=2{sin^4(π/8)+cos^4(π/8)}

=2[{sin^2(π/8)}^2+{cos^2(π/8)}^2]

=2[{sin^2(π/8)+cos^2(π/8)}-2.sin^2(π/8).cos^2(π/8)]

=2{1-2.sin^2(π/8).cos^2(π/8)}

=2[1-(1/2){2.sin(π/8).cos(π/8)}^2]

=2[1-(1/2){sin(2π/8)}^2]

=2{1-(1/2)(1/rt2)^2}=2{1-(1/4)}

=2(3/4)=3/2

=R.H.S

Hence cos^4(π/8)+cos^4(3π/8)+cos^4(5π/8)+cos^4(7π/8)=3/2 is proved

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