锐角α、β满足α+β=3/4π,求cos²α+cos²β+√2cosαcosβ的值1.锐角α、β满足α+β=3/4π,求cos²α+cos²β+√2cosαcosβ的值2.已知tan(α/2)=1/4tan(β/2),求证:tan[(β-α)/2]=3sinα/(5-3cosα)
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![锐角α、β满足α+β=3/4π,求cos²α+cos²β+√2cosαcosβ的值1.锐角α、β满足α+β=3/4π,求cos²α+cos²β+√2cosαcosβ的值2.已知tan(α/2)=1/4tan(β/2),求证:tan[(β-α)/2]=3sinα/(5-3cosα)](/uploads/image/z/13944468-12-8.jpg?t=%E9%94%90%E8%A7%92%CE%B1%E3%80%81%CE%B2%E6%BB%A1%E8%B6%B3%CE%B1%2B%CE%B2%3D3%2F4%CF%80%2C%E6%B1%82cos%26sup2%3B%CE%B1%2Bcos%26sup2%3B%CE%B2%2B%E2%88%9A2cos%CE%B1cos%CE%B2%E7%9A%84%E5%80%BC1.%E9%94%90%E8%A7%92%CE%B1%E3%80%81%CE%B2%E6%BB%A1%E8%B6%B3%CE%B1%2B%CE%B2%3D3%2F4%CF%80%EF%BC%8C%E6%B1%82cos%26sup2%3B%CE%B1%2Bcos%26sup2%3B%CE%B2%2B%E2%88%9A2cos%CE%B1cos%CE%B2%E7%9A%84%E5%80%BC2.%E5%B7%B2%E7%9F%A5tan%28%CE%B1%2F2%29%3D1%2F4tan%28%CE%B2%2F2%29%EF%BC%8C%E6%B1%82%E8%AF%81%EF%BC%9Atan%5B%28%CE%B2-%CE%B1%29%2F2%5D%3D3sin%CE%B1%2F%285-3cos%CE%B1%29)
锐角α、β满足α+β=3/4π,求cos²α+cos²β+√2cosαcosβ的值1.锐角α、β满足α+β=3/4π,求cos²α+cos²β+√2cosαcosβ的值2.已知tan(α/2)=1/4tan(β/2),求证:tan[(β-α)/2]=3sinα/(5-3cosα)
锐角α、β满足α+β=3/4π,求cos²α+cos²β+√2cosαcosβ的值
1.锐角α、β满足α+β=3/4π,求cos²α+cos²β+√2cosαcosβ的值
2.已知tan(α/2)=1/4tan(β/2),求证:tan[(β-α)/2]=3sinα/(5-3cosα)
锐角α、β满足α+β=3/4π,求cos²α+cos²β+√2cosαcosβ的值1.锐角α、β满足α+β=3/4π,求cos²α+cos²β+√2cosαcosβ的值2.已知tan(α/2)=1/4tan(β/2),求证:tan[(β-α)/2]=3sinα/(5-3cosα)
为了方便,我这里用a和b代替α、β
1、原式=cosa(cosa+√2/2cosb)+cosb(cosb+√2/2cosa)
a+b=3/4π
a=3/4π-b
cosa=cos(3/4π-b)=-√2/2cosb+√2/2sinb
代入
原式=(√2/2sinb-√2/2cosb)(√2/2sinb-√2/2cosb+√2/2cosb)+cosb(cosb+1/2sinb-1/2cosb)
=1/2sin²b-1/2sinbcosb+cosb(1/2cosb+1/2sinb)
=1/2sin²b-1/2sinbcosb+1/2cos²b+1/2sinbcosb
=1/2sin²b+1/2cos²b
=1/2
2、用万能公式,令a=α/2,b=β/2
tana=1/4tanb
tanb=4tana
左式=tan(b-a)=(tanb-tana)/(1+tanatanb)=(4tana-tana)/(1+4tan²a)=3tana/(1+4tan²a)
右式=[3×2tana/(1+tan²a)]/[5-3×(1-tan²a)/(1+tan²a)]=6tana/(5+5tan²a-3+3tan²a)
=6tana/(2+8tan²a)=3tana/(1+4tan²a)
左式=右式
证毕.