by Đẳng thức nào sau đây là sai?
A. ${{(\cos x+\sin x)}^{2}}+{{(\cos x-\sin x)}^{2}}=2,\forall x$.
B. ${{\tan }^{2}}x-{{\sin }^{2}}x={{\tan }^{2}}x{{\sin }^{2}}x,\forall x\ne {{90}^{0}}$.
C. ${{\sin }^{4}}x+{{\cos }^{4}}x=1-2{{\sin }^{2}}x{{\cos }^{2}}x,\forall x$.
D. ${{\sin }^{6}}x-{{\cos }^{6}}x=1-3{{\sin }^{2}}x{{\cos }^{2}}x,\forall x$.
Hướng dẫn
${{(\cos x+\sin x)}^{2}}+{{(\cos x-\sin x)}^{2}}={{\cos }^{2}}x+2\sin x. \cos x+{{\sin }^{2}}x+{{\cos }^{2}}x-2\sin x. \cos x+{{\sin }^{2}}x$
$=2\left( {{\cos }^{2}}x+{{\sin }^{2}}x \right)=2\Rightarrow A$đúng.
${{\tan }^{2}}x-{{\sin }^{2}}x=\frac{{{\sin }^{2}}x}{{{\cos }^{2}}x}-{{\sin }^{2}}x={{\sin }^{2}}x\left( \frac{1}{{{\cos }^{2}}x}-1 \right)=\frac{{{\sin }^{2}}x}{{{\cos }^{2}}x}. \left( 1-{{\cos }^{2}}x \right)={{\tan }^{2}}x. {{\sin }^{2}}x\Rightarrow B$đúng.
${{\sin }^{4}}x+{{\cos }^{4}}x={{\left( {{\sin }^{2}}x \right)}^{2}}+2{{\sin }^{2}}x. {{\cos }^{2}}x+{{\left( {{\cos }^{2}}x \right)}^{2}}-2{{\sin }^{2}}x. {{\cos }^{2}}x$
$ ={{\left( {{\sin }^{2}}x+{{\cos }^{2}}x \right)}^{2}}-2{{\sin }^{2}}x. {{\cos }^{2}}x=1-2{{\sin }^{2}}x. {{\cos }^{2}}x\Rightarrow C$ đúng.
Chọn đáp án D.
