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Lời giải:
\(A=(\sin ^2a)^3+(\cos ^2a)^3+3\sin ^2a\cos ^2a(\sin ^2a+\cos ^2a)\)
\(=(\sin ^2a+\cos ^2a)^3=1^3=1\)
\(B=(\cos ^2a+\sin ^2a-2\sin a\cos a)+(\cos ^2a+\sin ^2a+2\sin a\cos a)\)
\(=(1-2\sin a\cos a)+(1+2\sin a\cos a)=2\)
\(C=\frac{(\cos ^2a+\sin ^2a-2\sin a\cos a)-(\cos ^2a+\sin ^2a+2\sin a\cos a)}{\sin a\cos a}=\frac{(1-2\sin a\cos a)-(1+2\sin a\cos a)}{\sin a\cos a}\)
$=\frac{-4\sin a\cos a}{\sin a\cos a}=-4$
\(1+\sin^2\alpha+\cos^2\alpha=1+1=2\)
\(\sin^4\alpha+\cos^4\alpha+2\sin^2\alpha\cdot\cos^2\alpha\\ =\left(\sin^2\alpha\right)^2+2\sin^2\alpha\cdot\cos^2\alpha+\left(\cos^2\alpha\right)^2\\ =\left(\sin^2\alpha+\cos^2\alpha\right)^2\\ =1^2=1\)
\(\tan^2\alpha-\sin^2\alpha\cdot\tan^2\alpha\\ =\tan^2\alpha\left(1-\sin^2\alpha\right)\\ =\left(\frac{\sin\alpha}{\cos\alpha}\right)^2\cdot\cos^2\alpha\\ =\frac{\sin^2\alpha}{\cos^2\alpha}\cdot\cos^2\alpha\\ =\sin^2\alpha\)
\(\cos^2\alpha+\tan^2\alpha\cdot\cos^2\alpha\\ =\cos^2\alpha+\left(\frac{\sin\alpha}{\cos\alpha}\right)^2\cdot\cos^2\alpha\\ =\cos^2\alpha+\frac{\sin^2\alpha}{\cos^2\alpha}\cdot\cos^2\alpha\\ =\cos^2\alpha+\sin^2\alpha\\ =1\)
\(\tan^2\alpha\cdot\left(2\cos^2\alpha+\sin^2\alpha-1\right)\\ =\tan^2\alpha\cdot\left(2\cos^2\alpha+\sin^2\alpha-\sin^2\alpha-\cos^2\alpha\right)\\ =\tan^2\alpha\cdot\cos^2\alpha\\ =\frac{\sin^2\alpha}{\cos^2\alpha}\cdot\cos^2\alpha=\sin^2\alpha\)
a/ \(A=\frac{cot^2a-cos^2a}{cot^2a}-\frac{sina.cosa}{cota}\)
\(=\frac{\frac{cos^2a}{sin^2a}-cos^2a}{\frac{cos^2a}{sin^2a}}-\frac{sina.cosa}{\frac{cosa}{sina}}\)
\(=\left(1-sin^2a\right)-sin^2a=1\)
b/ \(B=\left(cosa-sina\right)^2+\left(cosa+sina\right)^2+cos^4a-sin^4a-2cos^2a\)
\(=cos^2a-2cosa.sina+sin^2a+cos^2a+2cosa.sina+sin^2a+\left(cos^2a+sin^2a\right)\left(cos^2a-sin^2a\right)-2cos^2a\)
\(=2+\left(cos^2a-sin^2a\right)-2cos^2a\)
\(=2-sin^2a-cos^2a=2-1=1\)
a) \(\cos^4\alpha-\sin^4\alpha=\left(\cos^2\alpha+\sin^2\alpha\right)\left(\cos^2\alpha-\sin^2\alpha\right)=\cos^2\alpha-\sin^2\alpha\)
\(2\cos^2\alpha-\left(\sin^2\alpha+\cos^2\alpha\right)=2\cos^2\alpha-1\)
b) \(\frac{\cos\alpha}{1-\sin\alpha}=\frac{1+\sin\alpha}{\cos\alpha}\)\(\Leftrightarrow\)\(\left(1-\sin\alpha\right)\left(1+\sin\alpha\right)=\cos^2\alpha\)
\(\Leftrightarrow\)\(1-\left(\sin^2\alpha+\cos^2\alpha\right)=0\)\(\Leftrightarrow\)\(1-1=0\) ( luôn đúng )
c) \(\frac{\left(\sin\alpha+\cos\alpha\right)^2-\left(\sin\alpha-\cos\alpha\right)^2}{\sin\alpha.\cos\alpha}=\frac{2\cos\alpha.2\sin\alpha}{\sin\alpha.\cos\alpha}=4\)
um, hình như câu b) chỗ 1-.... đó hơi sai nếu viết từ bước trên xuống á bạn!
mình nghĩ là: sau dấu bằng đầu tiên, sau đó là:
\(=cos^2\alpha=1-sin^2\alpha\)(luôn đúng)
CẢM ƠN bạn nhiều lắm luôn nha!!!!!