d la sai

a/ \(sin3x=sin\left(2x+x\right)=sin2xcosx+cos2x.sinx\)

\(=2sinxcos^2x+\left(1-2sin^2x\right)sinx=2sinx\left(1-sin^2x\right)+sinx-2sin^3x\)

\(=3sinx-4sin^3x\)

b/

\(tan2x+\frac{1}{cos2x}=\frac{sin2x}{cos2x}+\frac{1}{cos2x}=\frac{sin2x+1}{cos2x}=\frac{2sinxcosx+sin^2x+cos^2x}{cos^2x-sin^2x}\)

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

\(=\frac{cos^2x-sin^2x}{cos^2x+sin^2x-2sinxcosx}=\frac{1-2sin^2x}{1-sin2x}\)

c/

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

\(=\frac{2sinxcosx+2sinxcosx}{cos2x}=\frac{4sinxcosx}{cos2x}=\frac{2sin2x}{cos2x}=2tan2x\)

d/

\(\frac{sin2x}{1+cos2x}=\frac{2sinxcosx}{1+2cos^2x-1}=\frac{2sinxcosx}{2cos^2x}=\frac{sinx}{cosx}=tanx\)

e/

c/

\(\Leftrightarrow\frac{1}{2}-\frac{1}{2}cos2x+\frac{1}{2}-\frac{1}{2}cos6x=1-cos4x\)

\(\Leftrightarrow cos6x+cos2x-2cos4x=0\)

\(\Leftrightarrow2cos4x.cos2x-2cos4x=0\)

\(\Leftrightarrow2cos4x\left(cos2x-1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}cos4x=0\\cos2x=1\end{matrix}\right.\) \(\Leftrightarrow...\)

a/

\(\Leftrightarrow1+cos2x+cos3x+cosx=0\)

\(\Leftrightarrow2cos^2x+2cos2x.cosx=0\)

\(\Leftrightarrow2cosx\left(cosx+cos2x\right)=0\)

\(\Leftrightarrow2cosx\left(2cos^2x+cosx-1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}cosx=0\\cosx=-1\\cosx=\frac{1}{2}\end{matrix}\right.\) \(\Leftrightarrow...\)

b/

\(\Leftrightarrow2sin3x.cosx+sin3x=2cos3x.cosx+cos3x\)

\(\Leftrightarrow sin3x\left(2cosx+1\right)-cos3x\left(2cosx+1\right)=0\)

\(\Leftrightarrow\left(sin3x-cos3x\right)\left(2cosx+1\right)=0\)

\(\Leftrightarrow\sqrt{2}sin\left(3x-\frac{\pi}{4}\right)\left(2cosx+1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sin\left(3x-\frac{\pi}{4}\right)=0\\cosx=-\frac{1}{2}\end{matrix}\right.\) \(\Leftrightarrow...\)

\(sin^8x-cos^8x-4sin^6x+6sin^4x-4sin^2x\)

\(=sin^8x-\left(1-sin^2x\right)^4-4sin^6x+6sin^4x-4sin^2x\)

\(=sin^8x-\left(1-4sin^2x+6sin^4x-4sin^6x+sin^8x\right)-4sin^6x+6sin^4x-4sin^2x\)\(=-1\) (bạn chép nhầm đề)

b/ \(\frac{sin6x+sin2x+sin4x}{1+cos2x+cos4x}=\frac{2sin4x.cos2x+sin4x}{1+cos2x+2cos^22x-1}=\frac{sin4x\left(2cos2x+1\right)}{cos2x\left(2cos2x+1\right)}=\frac{sin4x}{cos2x}=\frac{2sin2x.cos2x}{cos2x}=2sin2x\)

c/ \(\frac{1+sin2x}{cosx+sinx}-\frac{1-tan^2\frac{x}{2}}{1+tan^2\frac{x}{2}}=\frac{sin^2x+cos^2x+2sinx.cosx}{cosx+sinx}-\left(1-tan^2\frac{x}{2}\right)cos^2\frac{x}{2}\)

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

d/ \(cos4x+4cos2x+3=2cos^22x-1+4cos2x+3\)

\(=2\left(cos^22x+2cos2x+1\right)=2\left(cos2x+1\right)^2=2\left(2cos^2x-1+1\right)^2=8cos^4x\)

e/

Cảm ơn ạ

Lời giải:

Ta có:

VT\(=\frac{1+\cot ^2x}{1-\cot ^2x}+\frac{\cos x}{\cos x-\sin x}=\frac{1+\left(\frac{\cos x}{\sin x}\right)^2}{1-\left(\frac{\cos x}{\sin x}\right)^2}+\frac{\cos x}{\cos x-\sin x}\)

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

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

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

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

Ta có đpcm.

\(\frac{sin2x}{cosx+cos3x}=\frac{2sinx.cosx}{2cos2x.cosx}=\frac{sinx}{cos2x}\)

\(\frac{1-cosx+cos2x}{sin2x-sinx}=\frac{1-cosx+2cos^2x-1}{2sinx.cosx-sinx}=\frac{cosx\left(2cosx-1\right)}{sinx\left(2cosx-1\right)}=\frac{cosx}{sinx}=cotx\)

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

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

\(C=\frac{sina.cosa\left(tana-cota\right)}{sina.cosa\left(tana+cota\right)}+cos2a=\frac{sin^2a-cos^2a}{sin^2a+cos^2a}+cos2a\)

\(=-cos2a+cos2a=0\)