If x=cos55degree ,y=cos65degree,z=cos175degree then xy+yz+zx=?
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We know that 2cosAcosB = cos(A + B) + cos(A-B).
= > (1/2)(cos(55 + 65) + cos(65 - 55)) + (1/2)(cos(65 + 175) + cos(175 - 65)) + (1/2)(cos(175 + 55)cos(175 - 55))
= > (1/2)(cos120+cos10) + (1/2)(cos240+cos110) + (1/2)(cos230 + cos120)
= > (1/2)(cos120) + (1/2)cos(10) + (1/2)cos240 + (1/2)cos(110) + (1/2)cos(230) + (1/2)cos120
= > (1/2)cos(180 - 60) + (1/2)cos(10) + (1/2)cos(180 + 60) + (1/2)cos(110) + (1/2)(cos230) - 1/2(cos(180 - 60))
= > (1/2)(-cos60) + (1/2)cos(10) + (1/2)cos(-cos60) + (1/2)cos110 + (1/2)cos230 - 1/2(-cos60)
= > (1/2)(-1/2) + (1/2)cos(10) + (1/2)(-1/2) + (1/2)cos(110) + (1/2)cos(230) - (1/2)(-1/2)
= > -1/4 + 1/2cos(10) - 1/4 + (1/2)cos(110) + (1/2)cos230 - 1/4
= > -(3/4) + (1/2)(cos10 + cos110 + cos230)
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