tam thoi cho ban dung

<=>(sinx+cosx-1)/(1-cosx+sinx+cosx-1)=(2cosx)/(sinx-cosx+1+2cosx)

<=>(sinx+cosx-1)/sinx=2cosx/(sinx+cosx+1)

x€(0;π/2)=> sinx ≠0; sinx+cosx+1≠0

<=>(sinx+cosx-1)(sinx+cosx+1)=2sinxcosx

<=>(sinx+cosx)^2-1=2sinxcosx

<=>(sin^2x+cos^2+2sinxcos)-1=2sinxcosx

<=>1+2sinxcosx-1=2sinxcosx

<=>2sinxcosx=2sinxcosx

moi bd <=>=> ban dung =>dpcm

ta có : \(0^o< x< 90^o\) \(\Rightarrow sinx-cosx+1>0\) và ta luôn có \(1-cosx>0\) \(\Rightarrow\) biểu thức trên được xác định

\(\Rightarrow\dfrac{sinx+cos-1}{1-cosx}=\dfrac{2cosx}{sinx-cos+1}\)

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

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

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

\(\Leftrightarrow sin^2x-cos^2x+2cosx-1=2cosx-2cos^2x\)

\(\Rightarrow2cosx-2cos^2x=2cosx-cos^2x\) \(\Rightarrow\left(đpcm\right)\)

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

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

\(\Leftrightarrow sin^2x-cos^2x+2cosx-1=2cosx-2cos^2x\)

\(\Leftrightarrow sin^2x+cos^2x-1=0\)

\(\Leftrightarrow1-1=0\) đúng

a) \(\left(sinx+cosx\right)^2=sin^2x+2sinxcosx+cos^2x\)\(=1+2sinxcosx\).
b) \(\left(sinx-cosx\right)^2=sin^2x-2sinxcosx+cos^2x\)\(=1-2sinxcosx\).
c) \(sin^4x+cos^4x=\left(sin^2x+cos^2x\right)^2-2sin^2xcos^2x\)
\(=1-2sin^2xcos^2x\).

\(B=\dfrac{1-4\sin^2x\cdot\cos^2x}{\sin^2x+2\sin x\cdot\cos x+\cos^2}+2\sin x\cdot\cos x\\ B=\dfrac{1-4\sin^2x\cdot\cos^2x}{2\sin x\cdot\cos x}+2\sin x\cdot\cos x\\ B=\dfrac{1-4\sin^2x\cdot\cos^2x+4\sin^2x\cdot\cos^2x}{2\sin x\cdot\cos x}=\dfrac{1}{2\sin x\cdot\cos x}\)

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

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

\(=\dfrac{2\left(cosx+1\right)}{sinx\left(cosx+1\right)}=\dfrac{2}{sinx}\)

à ra r. k cần giải nha m.n

Câu 1 đề sai, chắc chắn 1 trong 2 cái \(cot^2x\) phải có 1 cái là \(cos^2x\)

2.

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

\(=\dfrac{1-\left(sin^2x+cos^2x\right)}{cosx\left(1+sinx\right)}=\dfrac{1-1}{cosx\left(1+sinx\right)}=0\)

3.

\(\dfrac{tanx}{sinx}-\dfrac{sinx}{cotx}=\dfrac{tanx.cotx-sin^2x}{sinx.cotx}=\dfrac{1-sin^2x}{sinx.\dfrac{cosx}{sinx}}=\dfrac{cos^2x}{cosx}=cosx\)

4.

\(\dfrac{tanx}{1-tan^2x}.\dfrac{cot^2x-1}{cotx}=\dfrac{tanx}{1-tan^2x}.\dfrac{\dfrac{1}{tan^2x}-1}{\dfrac{1}{tanx}}=\dfrac{tanx}{1-tan^2x}.\dfrac{1-tan^2x}{tanx}=1\)

5.

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

\(=tan^2x+1+tan^2x=1+2tan^2x\)