关于三角函数的题目求证:cos6cos42cos66cos78=1/16 (全部是角度制)
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关于三角函数的题目
求证:cos6cos42cos66cos78=1/16 (全部是角度制)
求证:cos6cos42cos66cos78=1/16 (全部是角度制)
![关于三角函数的题目求证:cos6cos42cos66cos78=1/16 (全部是角度制)](/uploads/image/z/18831701-29-1.jpg?t=%E5%85%B3%E4%BA%8E%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0%E7%9A%84%E9%A2%98%E7%9B%AE%E6%B1%82%E8%AF%81%EF%BC%9Acos6cos42cos66cos78%3D1%2F16+%28%E5%85%A8%E9%83%A8%E6%98%AF%E8%A7%92%E5%BA%A6%E5%88%B6%EF%BC%89)
2*cosA*cosB = cos(A+B) + cos(A-B)
cosA + cosB = 2*[cos(A+B)/2]*cos[(A-B)/2]
cos6°cos42°cos66°cos78°
= (cos6°cos66°)*(cos42°cos78°)
= (cos72°+ cos60°)*(cos120°+ cos36°)/4
= (cos72°cos120°+ cos72°cos36°+ cos60°cos120°+ cos60°cos36°)/4
= [ ( cos36°- cos72°)/2 + cos72°cos36°- 1/4 )/4
= [ (cos36°- cos72°)/2 + (cos108°+ cos36°)/2 ]/4 - 1/16
= - 1/16
cosA + cosB = 2*[cos(A+B)/2]*cos[(A-B)/2]
cos6°cos42°cos66°cos78°
= (cos6°cos66°)*(cos42°cos78°)
= (cos72°+ cos60°)*(cos120°+ cos36°)/4
= (cos72°cos120°+ cos72°cos36°+ cos60°cos120°+ cos60°cos36°)/4
= [ ( cos36°- cos72°)/2 + cos72°cos36°- 1/4 )/4
= [ (cos36°- cos72°)/2 + (cos108°+ cos36°)/2 ]/4 - 1/16
= - 1/16