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\(=\left(sin^2\alpha\right)^3+\left(cos^2\alpha\right)^3+3sin^2\alpha-cos^2\alpha\)
\(=\left(sin^2\alpha+cos^2\alpha\right)\left(sin^4\alpha-sin^2\alpha.cos^2\alpha+cos^4\alpha\right)+3sin^2\alpha-cos^2\alpha\)
\(=sin^4\alpha-sin^2\alpha.cos^2\alpha+cos^4\alpha+3sin^2\alpha-cos^2\alpha\)
\(=sin^4\alpha+cos^4\alpha-sin^2\alpha.cos^2\alpha+3sin^2\alpha-cos^2\alpha\)
\(=\left(sin^2\alpha\right)^2+\left(cos^2\right)^2-sin^2\alpha.cos^2\alpha+3sin^2\alpha-cos^2\)
\(=1-2sin^2\alpha.cos^2\alpha-sin^2\alpha.cos^2\alpha+3sin^2\alpha-cos^2\alpha\)
\(=1-3sin^2\alpha.cos^2\alpha+3sin^2\alpha.cos^2\alpha-cos^2\alpha\)
\(=1-3sin^2\alpha.\left(1-sin^2\alpha\right)+3sin^2\alpha-\left(1-sin^2\alpha\right)\)
\(=1-3sin^2\alpha-sin^2\alpha+3sin^2\alpha-\left(1-sin^2\alpha\right)\)
\(1-3sin^2\alpha-sin^2\alpha+3sin^2\alpha-1+sin^2\alpha\)
\(=0\)
=(sin a+cos a)(sin^2.a-sina.cosa+cos^2a)+(sina+cosa)sina.cosa-cos a
=(sin a+cos a)(1-sina.cosa+sina.cosa)-cosa
=sina+cosa-cosa
=sina
Ta có:
\(sin^4a+cos^4a=\left(sin^2a+cos^2a\right)^2-2sin^2acos^2a=1-2sin^2a.cos^2a\)
Và:
\(sin^6a+cos^6a=\left(sin^2a+cos^2a\right)^3-3sin^2a.cos^2a.\left(sin^2a+cos^2a\right)\)
\(=1-3sin^2a.cos^2a\)
Do đó:
\(A=3\left(1-2sin^2a.cos^2a\right)-2\left(1-3sin^2a.cos^2a\right)=1\)
\(B=1-3sin^2.cos^2a+3sin^2a.cos^2a=1\)
\(=2\left[\left(sin^2a+cos^2a\right)^3-3\cdot sin^2a\cdot cos^2a\left(sin^2a+cos^2a\right)\right]-3\left[\left(sin^2a+cos^2a\right)^2-2\cdot sin^2a\cdot cos^2a\right]\)
\(=2\left[1-3sin^2acos^2a\right]-3\left[1-2sin^2acos^2a\right]\)
\(=2-6sin^2a\cdot cos^2a-3+6\cdot sin^2a\cdot cos^2a\)
=-1
Lời giải:
\(A=(\sin ^2a)^3+(\cos ^2a)^3+3\sin ^2a\cos ^2a(\sin ^2a+\cos ^2a)\)
\(=(\sin ^2a+\cos ^2a)^3=1^3=1\)
\(B=(\cos ^2a+\sin ^2a-2\sin a\cos a)+(\cos ^2a+\sin ^2a+2\sin a\cos a)\)
\(=(1-2\sin a\cos a)+(1+2\sin a\cos a)=2\)
\(C=\frac{(\cos ^2a+\sin ^2a-2\sin a\cos a)-(\cos ^2a+\sin ^2a+2\sin a\cos a)}{\sin a\cos a}=\frac{(1-2\sin a\cos a)-(1+2\sin a\cos a)}{\sin a\cos a}\)
$=\frac{-4\sin a\cos a}{\sin a\cos a}=-4$
\(\sin^4a.\left(3-2\sin^2a\right)+\cos^4a\left(3-2\cos^2a\right)\)
\(=3\sin^4a-2\sin^6a+3\cos^4a-2\cos^6a\)
\(=3\left(\sin^4a+\cos^4a\right)-2\left(\sin^6a+\cos^6a\right)\)
\(=3\left(\left(\sin^2a\right)^2+\left(\cos^2a\right)^2\right)-2\left(\left(\sin^2a\right)^3+\left(\cos^2a\right)^3\right)\)
\(=3.1-2\left(sin^2a+\cos^2a\right)\left(\sin^4-sin^2.\cos^2+\cos^4\right)\)
\(=3-2.1\left(\left(\sin^2a\right)^2+\left(\cos^2a\right)^2\right).\left(-\sin^2.\cos^2\right)\)
\(=3-2\left(-\sin^2.\cos^2\right)\)
\(A=sin^6a+cos^6a=\left(sin^2a+cos^2a\right)^3-3.sin^2a.cos^2a.\left(sin^2a+cos^2a\right)\)
\(=1-3sin^2a.cos^2a\)
Đặt \(x=sin^2a\) , \(y=cos^2a\) , thì ta có \(x^2+y^2=1\)
Áp dụng BĐT \(xy\le\frac{\left(x+y\right)^2}{4}\) , ta được :
\(sin^2a.cos^2a\le\frac{\left(sin^2a+cos^2a\right)^2}{4}=\frac{1}{4}\)
\(\Rightarrow A=1-3sin^2a.cos^2a\ge1-3.\frac{1}{4}=\frac{1}{4}\)
Đẳng thức xảy ra khi \(sin^2a=cos^2a\Leftrightarrow sina=cosa\Leftrightarrow a=45^o\)
Vậy........................................................................