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\(A=sin^6\alpha+cos^6\alpha+3sin^2\alpha-cos^2\alpha\)
\(=\left(sin^2\alpha\right)^3+\left(cos^2\alpha\right)^3+3sin^2\alpha-cos^2\alpha\)
\(=\left(sin^2\alpha+cos^2\alpha\right)\left(sin^4\alpha+cos^4\alpha-sin^2\alpha.cos^2\alpha\right)+3sin^2\alpha-cos^2\alpha\)
\(=sin^4\alpha+cos^4\alpha-sin^2\alpha.cos^2\alpha+3sin^2\alpha-cos^2\alpha\)
\(=\left(sin^2\alpha+cos^2\alpha\right)^2-2sin^2\alpha.cos^2\alpha-sin^2\alpha.cos^2\alpha+3sin^2\alpha-cos^2\alpha\)
\(1-3sin^2\alpha.cos^2\alpha+3sin^2\alpha-cos^2\alpha=3sin^2\alpha\left(1-cos^2\alpha\right)+\left(1-cos^2\alpha\right)\)
\(=\left(3sin^2\alpha+1\right).sin^2\alpha=0\)
=(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
\(\sin^6a+\cos^6a+3\sin^2a-\cos^2a\\ =\sin^6a+3\sin^2\cos^2\left(\sin^2a+\cos^2a\right)+\cos^6a-3\sin^2a\cos^2a\left(\sin^2a+\cos^2a\right)+3\sin^2a-\cos^2a\\ =\left(\sin^2a+\cos^2a\right)^3-3\sin^2a.\cos^2a.1+3\sin^2a-cos^2a\\ =1^3-\cos^2a+3\sin^2a-3\sin^2\cos^2\\ =\left(1-\cos^2a\right)\left(3\sin^2a+1\right)\)
Đặt \(\sin^2\alpha=x\Rightarrow\cos^2\alpha=1-\sin^2\alpha\)
\(A=x^3+\left(1-x\right)^3+3x-\left(1-x\right)=x^3+1-3x+3x^2-x^3+3x-1+x=3x^2+x\)
Vậy \(A=3\sin^4\alpha+\sin^2\alpha\). NHỚ NHA!
a) ta có : \(A=\left(sin\alpha+cos\alpha\right)^2+\left(sin\alpha-cos\alpha\right)^2\)
\(\Leftrightarrow A=sin^2\alpha+2sin\alpha.cos\alpha+cos^2\alpha+sin^2\alpha-2sin\alpha.cos\alpha+cos^2\alpha\)
\(\Leftrightarrow A=2\left(sin^2\alpha+cos^2\alpha\right)=2.1=2\) (không phụ thuộc vào \(\alpha\))
\(\Rightarrow\left(đpcm\right)\)
\(B=sin^6\alpha+cos^6\alpha+3sin^2\alpha.cos^2\alpha\)
\(\Leftrightarrow B=\left(sin^2\alpha+cos^2\alpha\right)^3-3sin^2\alpha.cos^2\alpha\left(sin^2\alpha+cos^2\alpha\right)+3sin^2\alpha.cos^2\alpha\)
\(\Leftrightarrow B=\left(sin^2\alpha+cos^2\alpha\right)^3-3sin^2\alpha.cos^2\alpha+3sin^2\alpha.cos^2\alpha\)
\(\Leftrightarrow B=\left(sin^2\alpha+cos^2\alpha\right)^3=1^3=1\) (không phụ thuộc vào \(\alpha\) ) \(\Rightarrow\left(đpcm\right)\)
a/A = sin2 + 2. sin.cos + cos2 + sin2 -2cos.sin + cos2= 2
Tớ không biết ghi anpha nên .. 
\(=\left(sin^2\alpha\right)^3+\left(cos^2\alpha\right)^3+3sin^2\alpha-cos^2\alpha\)
\(=\left(sin^2\alpha+cos^2\alpha\right)\left(sin^4\alpha-sin^2\alpha.cos^2\alpha+cos^4\alpha\right)+3sin^2\alpha-cos^2\alpha\)
\(=sin^4\alpha-sin^2\alpha.cos^2\alpha+cos^4\alpha+3sin^2\alpha-cos^2\alpha\)
\(=sin^4\alpha+cos^4\alpha-sin^2\alpha.cos^2\alpha+3sin^2\alpha-cos^2\alpha\)
\(=\left(sin^2\alpha\right)^2+\left(cos^2\right)^2-sin^2\alpha.cos^2\alpha+3sin^2\alpha-cos^2\)
\(=1-2sin^2\alpha.cos^2\alpha-sin^2\alpha.cos^2\alpha+3sin^2\alpha-cos^2\alpha\)
\(=1-3sin^2\alpha.cos^2\alpha+3sin^2\alpha.cos^2\alpha-cos^2\alpha\)
\(=1-3sin^2\alpha.\left(1-sin^2\alpha\right)+3sin^2\alpha-\left(1-sin^2\alpha\right)\)
\(=1-3sin^2\alpha-sin^2\alpha+3sin^2\alpha-\left(1-sin^2\alpha\right)\)
\(1-3sin^2\alpha-sin^2\alpha+3sin^2\alpha-1+sin^2\alpha\)
\(=0\)