Rút gọn các biểu thức sau:

a) ;$A=\frac{1-2{\text{sin}}^{2}x}{1+\text{sin}2x}-\frac{1-\text{tan}x}{1+\text{tan}x}$

a) $A=\frac{1-2{\text{sin}}^{2}x}{1+\text{sin}2x}-\frac{1-\text{tan}x}{1+\text{tan}x}$

$=\frac{{\sin }^{2}x+co{s}^{2}x-2{\text{sin}}^{2}x}{{\sin }^{2}x+co{s}^{2}x+2\sin x\cos x}-\frac{1-\frac{\mathrm{s}\text{inx}}{\cos x}}{1+\frac{\mathrm{s}\text{inx}}{\cos x}}$

$=\frac{co{s}^{2}x-{\text{sin}}^{2}x}{{\left(\sin x+\cos x\right)}^2}-\frac{\cos x-\mathrm{s}\text{inx}}{\cos x+\mathrm{s}\text{inx}}$

$=\frac{\left(cosx-\text{sin}x\right)\left(\cos x+\sin x\right)}{{\left(\sin x+\cos x\right)}^2}-\frac{\cos x-\mathrm{s}\text{inx}}{\cos x+\mathrm{s}\text{inx}}$

$=\frac{cosx-\text{sin}x}{\sin x+\cos x}-\frac{\cos x-\mathrm{s}\text{inx}}{\cos x+\mathrm{s}\text{inx}}=0$.