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Answer 1 · 2017-11-21T14:41:12

#(sqrt(6)-sqrt(2))/4~~0.258819045#

#(5pi)/12=(2pi)/12+(3pi)/12=pi/4+pi/6#

Using the identity:

#color(red)(cos(A+B)=cosAcosB-sinAsinB)#

#cos(pi/4+pi/6)=cos(pi/4)cos(pi/6)-sin(pi/4)sin(pi/6)#

#cos(pi/4)=sqrt(2)/2#

#cos(pi/6)=sqrt(3)/2#

#sin(pi/4)=sqrt(2)/2#

#sin(pi/6)=1/2#

#:.#

#sqrt(2)/2*sqrt(3)/2-sqrt(2)/2*1/2=sqrt(2)/2(sqrt(3)/2-1/2)->=sqrt(2)/2((sqrt(3)-1)/2)=color(blue)((sqrt(6)-sqrt(2))/4~~0.258819045)#