证明sin^2(x)+cos^2(x+30)+sin(x)cos(x+30)=3/4

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证明sin^2(x)+cos^2(x+30)+sin(x)cos(x+30)=3/4
证明sin^2(x)+cos^2(x+30)+sin(x)cos(x+30)=3/4

证明sin^2(x)+cos^2(x+30)+sin(x)cos(x+30)=3/4
sin^2(x)+cos^2(x+30)+sin(x)cos(x+30）
=sin^2(x)+cos(x+30)[cos(x+30) +sinx]
=sin^2(x) + cos(x+30)(cosxcos30 -sinxsin30 +sinx)
=sin^2(x) + cos(x+30)(cosxcos30+1/2 *sinx)
=sin^2(x) + cos(x+30)(cosxcos30+sinxsin30)
=sin^2(x) +(cosxcos30-sinxsin30)(cosxcos30+sinxsin30)
=sin^2(x) +cos^2(x)*(cos30)^2 -(sin30)^2*sin^2(x)
=3/4 *cos^2(x)+sin^2(x)-1/4*sin^2(x)
=3/4 *cos^2(x)+3/4*sin^2(x)=3/4

∵sin^2(x)+cos^2(x+30)+sin(x)cos(x+30)
=sin²x+(cosxcos30-sinxsin30)²+sinx(cosxcos30-sinxsin30)
=sin²x+(√3cosx/2-sinx/2)²+sinx(√3cosx/2-sinx/2)
=s...

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∵sin^2(x)+cos^2(x+30)+sin(x)cos(x+30)
=sin²x+(cosxcos30-sinxsin30)²+sinx(cosxcos30-sinxsin30)
=sin²x+(√3cosx/2-sinx/2)²+sinx(√3cosx/2-sinx/2)
=sin²x+3cos²x/4-√3sinxcosx/2+sin²x/4+√3sinxcosx/2-sin²x/2
=3sin²x/4+3cos²x/4
=3(sin²x+cos²x)/4
=3/4
∴原命题成立。

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