The value of that <span>(√5 + 1)/8 and We can prove it like this: </span>cos 2A= 2cos^2 A-1=1-2sin^2 A <span>so cos^2 48 – sin^2 12 </span> <span>=(1/2)(1+cos 96) - (1/2)(1-cos24) </span> <span>=(1/2)(cos 96 +cos 24) and using cos(A)+cosB)=2cos((A+B)/2)cos((A-B)/2) </span> <span>=cos60cos36 </span> <span>=(1/2)cos36 </span> <span>and you have to find cos36. </span> <span>Suppose 5x=180, then </span> <span>cos(5x)=cos(180) then x=-180,-108, -36, 36, 108 </span> <span>Also 3x=180-2x </span> <span>so cos(3x)=cos(180-2x)=-cos2x </span> <span>so 4cos^3(x) -3cos(x)=1 - 2cos^2(x) </span> <span>giving 4c^3+2c^2 - 3c-1=0 where c=cosx </span> <span>c=-1 is a root and factorizing gives </span> <span>(c+1)(4c^2-2c-1)=0 </span> <span>so 4c^-2c-1=0 giving </span> <span>c=(2±√20)/8=(1±√5)/4 and these have values cos(36) and cos(108) </span> <span>the positive root is therefore cos(36)=(1+√5)/4 </span> <span>and the required value (1/2)cos(36)=(1+√5)/8 </span>I think this can be very useful
