\(tanx+\frac{cosx}{1+sinx}=\frac{sinx}{cosx}+\frac{cosx}{1+sinx}=\frac{sinx+sin^2x+cos^2x}{\left(1+sinx\right)cosx}=\frac{1+sinx}{\left(1+sinx\right)cosx}=\frac{1}{cosx}\)

\(tanx+\frac{cosx}{1+sinx}\)

\(=\frac{cosx}{1+sinx}+\frac{sinx}{cosx}\)

\(=\frac{cos^2x}{cosx.\left(sinx+1\right)}+\frac{sinx.\left(sinx+1\right)}{cosx.\left(sinx+1\right)}\)

\(=\frac{cos^2x+sinx.\left(sinx+1\right)}{cosx.\left(sinx+1\right)}\)

\(=\frac{1-sin^2x+\left(1+sinx\right)sinx}{\left(1+sinx\right).cosx}\)

\(=\frac{sinx+1}{cosx.\left(sinx+1\right)}\)

\(=\frac{1}{cosx}\)

\(A=\frac{sin^2x+cos^2x+2sinx.cosx-1}{\frac{cosx}{sinx}-sinx.cosx}=\frac{2sinx^2x.cosx}{cosx-sin^2x.cosx}=\frac{2sin^2x.cosx}{cosx\left(1-sin^2x\right)}\)

\(=\frac{2sin^2x}{1-sin^2x}=\frac{2sin^2x}{cos^2x}=2tan^2x\)

\(N=\left(\frac{sinx+\frac{sinx}{cosx}}{cosx+1}\right)^2+1=\left(\frac{sinx.cosx+sinx}{cosx\left(cosx+1\right)}\right)^2+1\)

\(=\left(\frac{sinx\left(cosx+1\right)}{cosx\left(cosx+1\right)}\right)^2+1=tan^2x+1=\frac{1}{cos^2x}\)

\(P=\frac{1-sin^2x.cos^2x}{cos^2x}-cos^2x=\frac{1}{cos^2x}-sin^2x-cos^2x\)

\(=1+tan^2x-\left(sin^2x+cos^2x\right)=1+tan^2x-1=tan^2x\)

\(M=\frac{2cos^2x-1}{sinx+cosx}=\frac{2cos^2x-\left(sin^2x+cos^2x\right)}{sinx+cosx}=\frac{cos^2x-sin^2x}{sinx+cosx}\)

\(\frac{\left(cosx-sinx\right)\left(cosx+sinx\right)}{sinx+cosx}=cosx-sinx\)

a) (1 -cosx)(1+cosx)

=\(\left(1-cos^2x\right)-sin^2x\)

=\(sin^2x-sin^2x\)

=0

b) tan\(^2x\)(2cos\(^2x\)+sin\(^2x\)-1) +cos\(^2x\)

\(=tan^2x\left(cos^2x+cos^2x+sin^2x-1\right)\)+\(cos^2x\)

=\(tan^2x\left(cos^2x+1-1\right)+cós^2x\)

\(=tan^2x.cos^2x+cos^2x \)

=\(\dfrac{sin^2x}{cos^2x}.cos^2x+cos^2x\)

=\(sin^2x+cos^2x\)

=1

\(sinx-cos\left(\pi-x\right)=-\frac{1}{2}\)

\(\Leftrightarrow sinx+cosx=-\frac{1}{2}\)

\(\Rightarrow\left(sinx+cosx\right)^2=\frac{1}{4}\)

\(\Rightarrow sin^2x+cos^2x+2sinx.cosx=\frac{1}{4}\)

\(\Rightarrow1+2sinx.cosx=\frac{1}{4}\Rightarrow sinx.cosx=-\frac{3}{8}\)

\(T=\frac{1}{sinx}+\frac{1}{cosx}=\frac{sinx+cosx}{sinx.cosx}=\frac{-\frac{1}{4}}{-\frac{3}{8}}=\frac{2}{3}\)

đề bài đầy đủ: rút gọn các biểu thức lượng giác sau trên điều kiện xác định của chúng:

\(\frac{sin^2x}{cosx+cosx.\frac{sinx}{cosx}}-\frac{cos^2x}{sinx+sinx.\frac{cosx}{sinx}}=\frac{sin^2x}{sinx+cosx}-\frac{cos^2x}{sinx+cosx}=\frac{sin^2x-cos^2x}{sinx+cosx}\)

\(=\frac{\left(sinx+cosx\right)\left(sinx-cosx\right)}{sinx+cosx}=sinx-cosx\)

\(\left(\frac{sinx}{cosx}+\frac{cosx}{1+sinx}\right)\left(\frac{cosx}{sinx}+\frac{sinx}{1+cosx}\right)=\left(\frac{sinx+sin^2x+cos^2x}{cosx\left(1+sinx\right)}\right)\left(\frac{cosx+cos^2x+sin^2x}{sinx\left(1+cosx\right)}\right)\)

\(=\left(\frac{sinx+1}{cosx\left(1+sinx\right)}\right)\left(\frac{cosx+1}{sinx\left(1+cosx\right)}\right)=\frac{1}{sinx.cosx}\)

\(=2sin2x.cosx-2sinx.cosx+2cosx-2cos^2x\)

\(=2cosx\left(sin2x+1\right)-2cosx\left(sinx+cosx\right)\)

\(=2cosx\left(2sinx.cosx+sin^2x+cos^2x\right)-2cosx\left(sinx+cosx\right)\)

\(=2cosx\left(sinx+cosx\right)^2-2cosx\left(sinx+cosx\right)\)

\(=2cosx\left(sinx+cosx\right)\left(sinx+cosx-1\right)\)