Giá trị biểu thức $ B=\dfrac{{{\cos }^{2}}x-{{\sin }^{2}}y}{{{\sin }^{2}}x.{{\sin }^{2}}y}-{{\cot }^{2}}x.{{\cot }^{2}}y $ bằng
Giá trị biểu thức $ B=\dfrac{{{\cos }^{2}}x-{{\sin }^{2}}y}{{{\sin }^{2}}x.{{\sin }^{2}}y}-{{\cot }^{2}}x.{{\cot }^{2}}y $ bằng
By Jade
Ta có $ B=\dfrac{{{\cos }^{2}}x-{{\sin }^{2}}y}{{{\sin }^{2}}x.{{\sin }^{2}}y}-{{\cot }^{2}}x.{{\cot }^{2}}y=\dfrac{{{\cos }^{2}}x-{{\sin }^{2}}y}{{{\sin }^{2}}x{{\sin }^{2}}y}-\dfrac{{{\cos }^{2}}x.{{\cos }^{2}}y}{{{\sin }^{2}}x.{{\sin }^{2}}y} $
$ =\dfrac{{{\cos }^{2}}x\left( 1-{{\cos }^{2}}y \right)-{{\sin }^{2}}y}{{{\sin }^{2}}x{{\sin }^{2}}y}=\dfrac{{{\cos }^{2}}x{{\sin }^{2}}y-{{\sin }^{2}}y}{{{\sin }^{2}}x{{\sin }^{2}}y}=\dfrac{{{\sin }^{2}}y\left( {{\cos }^{2}}x-1 \right)}{\left( 1-{{\cos }^{2}}x \right){{\sin }^{2}}y}=-1 $ .
B=$\frac{cot²x}{sin²y}$ -$\frac{1}{sin²x}$ -cot²x.cot²y
=cot²x($\frac{1}{sin²y}$ -cot²y) -$\frac{1}{sin²x}$
=cot²x($\frac{1}{sin²y}$-$\frac{cos²y}{sin²y}$ ) -$\frac{1}{sin²x}$
=cot²x($\frac{1-cos²y}{sin²y}$ )-$\frac{1}{sin²x}$
= cot²x-$\frac{1}{sin²x}$=$\frac{cos²x}{sin²x}$ -$\frac{1}{sin²x}$=$\frac{-sin²x}{sin²x}$ =-1