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Question Number 119109 by bagjagunawan last updated on 22/Oct/20

Answered by 1549442205PVT last updated on 22/Oct/20

P=((1/2)+cos(π/(20)))((1/2)+cos((3π)/(20)))((1/2)+cos((9π)/(20)))((1/2)+cos((27π)/(20)))  =((1/2)+cos9)((1/2)+cos81)((1/2)+cos27)((1/2)−cos63)  =((1/4)+((cos9+cos81)/2)+cos9cos81)  ×((1/4)+((cos27−cos63)/2)−cos27cos63)  =((1/4)+cos45cos36+((cos90+cos72)/2))  ×((1/4)+sin18sin45−((cos90+cos36)/2))  =((1/4)+((√2)/2)cos36+((sin18)/2))((1/4)+((√2)/2)sin18−((cos36)/2))(1)  •sin36=cos54⇔2sin18cos18=4cos^3 18−3cos18  ⇔2sin18=4cos^2 18−3=4(1−sin^2 18)−3  ⇒4sin^2 18+2sin18−1=0  sin18=(((√5)−1)/4)⇒cos36=1−2sin^2 18  =1−((3−(√5))/4)=(((√5)+1)/4).Replace into (1)we  get P=((1/4)+((√2)/2).(((√5)+1)/4)+(((√5)−1)/8))  ×((1/4)+((√2)/2).(((√5)−1)/4)−(((√5)+1)/8))  =(((√5)+1+(√2)((√5)+1))/8)×(((√2)((√5)−1)−(√5)+1)/8)  =(((√(10))+1+(√5)+(√2))/8).(((√(10))+1−((√5)+(√2)))/8)  =((11+2(√(10))−(7+2(√(10))))/(64))=(1/(16))

P=(12+cosπ20)(12+cos3π20)(12+cos9π20)(12+cos27π20)=(12+cos9)(12+cos81)(12+cos27)(12cos63)=(14+cos9+cos812+cos9cos81)×(14+cos27cos632cos27cos63)=(14+cos45cos36+cos90+cos722)×(14+sin18sin45cos90+cos362)=(14+22cos36+sin182)(14+22sin18cos362)(1)sin36=cos542sin18cos18=4cos3183cos182sin18=4cos2183=4(1sin218)34sin218+2sin181=0sin18=514cos36=12sin218=1354=5+14.Replaceinto(1)wegetP=(14+22.5+14+518)×(14+22.5145+18)=5+1+2(5+1)8×2(51)5+18=10+1+5+28.10+1(5+2)8=11+210(7+210)64=116

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