a: \(\left(1-cosx\right)\left(1+cosx\right)=1^2-cos^2x=sin^2x\)

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

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

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

c: \(sin^4x+cos^4x+2\cdot cos^2x\cdot sin^2x\)

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

\(=1^2=1\)

\(A=\left(\sin\alpha+\cos\alpha+\sin\alpha-\cos\alpha\right)^2-2\left(\sin\alpha+\cos\alpha\right)\left(\sin\alpha-\cos\alpha\right)\)

\(=4\sin^2\alpha-2\sin^2\alpha+2\cos^2\alpha=2\left(\sin^2\alpha+\cos^2\alpha\right)=2\)

\(B=\sin^4\alpha+\cos^4\alpha+2\sin^2\alpha.\cos^2\alpha\left(\sin^2\alpha+\cos^2\alpha\right)=\sin^4\alpha+\cos^4\alpha+2\sin^2\alpha.\cos^2\alpha\)

\(=\left(\sin^2\alpha+\cos^2\alpha\right)^2-1=0\)

\(C=3\left(\sin^4\alpha+\cos^4\alpha\right)-2\sin^2\alpha.\cos^2\alpha\left(\sin^2\alpha+\cos^2\alpha\right)=3\left(\sin^4\alpha+\cos^4\alpha\right)-2\sin^2\alpha.\cos^2\alpha\)

\(=3\left(\sin^2\alpha+\cos^2\alpha-\frac{1}{9}\right)^2-\frac{1}{9}=\frac{61}{27}\)

Lời giải:

a)

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

b)

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

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

c)

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

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

d)

\(\frac{\cos ^2x-\sin ^2x}{\cot ^2x-\tan ^2x}=\frac{\cos ^2x-\sin ^2x}{\frac{\cos ^2x}{\sin ^2x}-\frac{\sin ^2x}{\cos ^2x}}=\frac{\sin ^2x\cos ^2x(\cos ^2x-\sin ^2x)}{\cos ^4x-\sin ^4x}\)

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

e)

\(1-\cot ^4x=1-\frac{\cos ^4x}{\sin ^4x}=\frac{\sin ^4x-\cos ^4x}{\sin ^4x}=\frac{(\sin ^2x-\cos ^2x)(\sin ^2x+\cos ^2x)}{\sin ^4x}\)

\(=\frac{\sin ^2x-\cos ^2x}{\sin ^4x}=\frac{\sin ^2x-(1-\sin ^2x)}{\sin ^4x}=\frac{2\sin ^2x-1}{\sin ^4x}=\frac{2}{\sin ^2x}-\frac{1}{\sin ^4x}\)

Ta có ddpcm.

a/

\(\Leftrightarrow cos2x=sin3x\)

\(\Leftrightarrow cos2x=cos\left(\frac{\pi}{2}-3x\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}2x=\frac{\pi}{2}-3x+k2\pi\\2x=3x-\frac{\pi}{2}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{10}+\frac{k2\pi}{5}\\x=\frac{\pi}{2}+k2\pi\end{matrix}\right.\)

b/

\(\Leftrightarrow\left(sinx-1\right)\left(2sinx+1\right)\left(sin^2x-2sinx-1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=1\\sinx=-\frac{1}{2}\\sinx=1-\sqrt{2}\end{matrix}\right.\) \(\Leftrightarrow x=...\)

c/

\(\Leftrightarrow\left(sin^2x+cos^2x\right)^3-3sin^2x.cos^2x\left(sin^2x+cos^2x\right)=1+cos\left(\frac{\pi}{2}-2x\right)\)

\(\Leftrightarrow1-3sin^2x.cos^2x=1+sin2x\)

\(\Leftrightarrow-\frac{3}{4}sin^22x=sin2x\)

\(\Leftrightarrow3sin^22x+4sin2x=0\)

\(\Leftrightarrow sin2x\left(3sin2x+4\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sin2x=0\\sin2x=-\frac{4}{3}\left(l\right)\end{matrix}\right.\)

\(\Rightarrow x=\frac{k\pi}{2}\)

a) 1- \(sin^2\alpha\)= \(cos^2\alpha\)

b) (\(1-cos\alpha\))(\(1+cos\alpha\)) = 1 - cos²\(\alpha\) = sin²\(\alpha\)

c) 1 + cos²\(\alpha\) + sin²\(\alpha\) = \(1+1=2\)

d) sin\(\alpha\) - sin\(\alpha.cos^2\alpha\)

= \(sin\alpha\left(1-cos^2\alpha\right)=sin\alpha.sin^2\alpha=sin^3\alpha\)

e) \(sin^4\alpha+cos^4\alpha+2sin^2\alpha.cos^2\alpha\)

= \(\left(sin^2\alpha\right)^2+2sin^2\alpha.cos^2\alpha+\left(cos^2\alpha\right)^2\)

= \(\left(sin^2\alpha+cos^2\alpha\right)^2=1^2=1\)

f) \(tan^2\alpha-sin^2\alpha.tan^2\alpha\)

= \(tan^2\alpha\left(1-sin^2\alpha\right)=tan^2\alpha.cos^2\alpha=sin^2\alpha\)

g) \(cos^2\alpha+tan^2\alpha.cos^2\alpha\)

= \(cos^2\alpha\left(1+tan^2\alpha\right)=cos^2\alpha.\dfrac{1}{cos^2\alpha}=1\)

h) \(tan^2\alpha\left(2cos^2\alpha+sin^2\alpha-1\right)\)

= \(tan^2\alpha\left[cos^2\alpha+\left(cos^2\alpha+sin^2\alpha\right)-1\right]\)

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

= \(tan^2\alpha.cos^2\alpha=sin^2\alpha\)