#Sin^6(π/8)+sin^6(3π/8)+sin^6(5π/8)+sin^6(7π/8)=5/4#?

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Answer 1 · 2018-06-16T12:14:35

#LHS=Sin^6(π/8)+sin^6(3π/8)+sin^6(5π/8)+sin^6(7π/8)#

#=Sin^6(π/8)+sin^6((3π)/8)+sin^6(pi-(3π)/8)+sin^6(pi-π/8)#

#=Sin^6(π/8)+sin^6((3π)/8)+sin^6((3π)/8)+sin^6(π/8)#

#=2(Sin^6(π/8)+sin^6((3π)/8))#

#=2(Sin^6(π/8)+sin^6(pi/2-π/8))#

#=2(Sin^6(π/8)+cos^6(π/8))#

#=2[(Sin^2(π/8)+cos^2(π/8))^3-3(Sin^2(π/8)cos^2(π/8))(Sin^2(π/8)+cos^2(π/8))]#

#=2[1^3-3/4(4Sin^2(π/8)cos^2(π/8))*1]#

#=2[1^3-3/4(2Sin(π/8)cos(π/8))^2]#

#=2[1^3-3/4*Sin^2(π/4)]#

#=2[1^3-3/4*(1/sqrt2)^2]#

#=2[1-3/8]#

#=2*5/8=5/4=RHS#