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a)
\(\sin ^4a-\cos ^4a+1=(\sin ^2a-\cos ^2a)(\sin ^2a+\cos^2a)+1\)
\(=(\sin ^2a-\cos ^2a).1+1=\sin ^2a-\cos ^2a+\sin ^2a+\cos ^2a\)
\(=2\sin ^2a\)
b) \(\sin ^2a+2\cos ^2a-1=(\sin ^2a+\cos^2a)+\cos ^2a-1\)
\(=1+\cos ^2a-1=\cos ^2a\)
\(\Rightarrow \frac{\sin ^2a+2\cos ^2a-1}{\cot ^2a}=\frac{\cos ^2a}{\cot ^2a}=\frac{\cos ^2a}{\frac{\cos ^2a}{\sin ^2a}}=\sin ^2a\)
c)
\(\frac{1-\sin ^2a\cos ^2a}{\cos ^2a}-\cos ^2a=\frac{1}{\cos ^2a}-\sin ^2a-\cos ^2a\)
\(=\frac{1}{\cos ^2a}-(\sin ^2a+\cos ^2a)=\frac{1}{\cos ^2a}-1\)
\(=\frac{1-\cos ^2a}{\cos ^2a}=\frac{\sin ^2a}{\cos ^2a}=\tan ^2a\)
d)
\(\frac{\sin ^2a-\tan ^2a}{\cos ^2a-\cot ^2a}=\frac{\sin ^2a-\frac{\sin ^2a}{\cos ^2a}}{\cos ^2a-\frac{\cos ^2a}{\sin ^2a}}\) \(=\frac{\sin ^2a(1-\frac{1}{\cos ^2a})}{\cos ^2a(1-\frac{1}{\sin ^2a})}\)
\(=\frac{\sin ^2a.\frac{\cos ^2a-1}{\cos ^2a}}{\cos ^2a.\frac{\sin ^2a-1}{\sin ^2a}}\) \(=\frac{\sin ^2a.\frac{-\sin ^2a}{\cos ^2a}}{\cos ^2a.\frac{-\cos ^2a}{\sin ^2a}}=\frac{\sin ^6a}{\cos ^6a}=\tan ^6a\)
f)
\(\frac{(\sin a+\cos a)^2-1}{\cot a-\sin a\cos a}=\frac{\sin ^2a+\cos ^2a+2\sin a\cos a-1}{\frac{\cos a}{\sin a}-\sin a\cos a}\)
\(=\sin a.\frac{1+2\sin a\cos a-1}{\cos a-\cos a\sin ^2a}\)
\(=\sin a. \frac{2\sin a\cos a}{\cos a(1-\sin ^2a)}=\sin a. \frac{2\sin a\cos a}{\cos a. \cos^2 a}=\frac{2\sin ^2a}{\cos ^2a}=2\tan ^2a\)
a/ \(\dfrac{\sin x+\cos x-1}{1-\cos x}=\dfrac{2\cos x}{\sin x-\cos x+1}\)
\(\Leftrightarrow-2\cos^2x+2\cos x-2\cos x+2\cos^2x=0\)
\(\Leftrightarrow0=0\) (đúng)
\(\RightarrowĐPCM\)
b/ \(\tan a.\tan b=\dfrac{\tan a+\tan b}{\cot a+\cot b}\)
\(\Leftrightarrow\tan a.\tan b.\left(\cot a+\cot b\right)=\tan a+\tan b\)
\(\Leftrightarrow\tan a.\tan b.\cot a+\tan a.\tan b.\cot b=\tan a+\tan b\)
\(\Leftrightarrow\tan b+\tan a=\tan a+\tan b\) (đúng)
\(\RightarrowĐPCM\)
\(VT=\dfrac{1+2cos^2\dfrac{a}{2}-1-2sin\dfrac{a}{2}cos\dfrac{a}{2}}{1-\left(1-2sin^2\dfrac{a}{2}\right)-2sin\dfrac{a}{2}cos\dfrac{a}{2}}=\dfrac{2cos^2\dfrac{a}{2}-2sin\dfrac{a}{2}cos\dfrac{a}{2}}{2sin^2\dfrac{a}{2}-2sin\dfrac{a}{2}cos\dfrac{a}{2}}\)
\(=\dfrac{2cos\dfrac{a}{2}\left(cos\dfrac{a}{2}-sin\dfrac{a}{2}\right)}{2sin\dfrac{a}{2}\left(sin\dfrac{a}{2}-cos\dfrac{a}{2}\right)}\)
\(=-\dfrac{cos\dfrac{a}{2}}{sin\dfrac{a}{2}}=-cot\dfrac{a}{2}=VP\\ \Rightarrowđpcm\)
2.
\(\text{VP}=\frac{1}{32}(2+\cos 2x-2\cos 4x-\cos 6x)\)
\(=\frac{1}{32}[2+\cos 2x-2(2\cos ^22x-1)-(4\cos ^32x-3\cos 2x)]\)
\(=\frac{1}{8}(-\cos ^32x-\cos ^22x+\cos 2x+1)=\frac{1}{8}(\cos 2x+1)(1-\cos ^22x)=\frac{1}{8}(\cos 2x+1)\sin ^22x\) (1)
\(\text{VT}=\sin ^2x\cos ^4x=\frac{1}{8}.(2\sin x\cos x)^2.2\cos ^2x=\frac{1}{8}\sin ^22x.(\cos 2x+1)(2)\)
Từ $(1);(2)$ ta có đpcm.
1.
\(\sin ^8x-\cos ^8x=(\sin ^4x+\cos ^4x)(\sin ^4x-\cos ^4x)\)
\(=[(\sin ^2x+\cos ^2x)^2-2\sin ^2x\cos ^2x](\sin ^2x+\cos ^2x)(\sin ^2x-\cos ^2x)\)
\(=(1-2\sin ^2x\cos ^2x)(\sin ^2x-\cos ^2x)\)
\(=(1-\frac{\sin ^22x}{2})(-\cos 2x)=-\frac{(2-\sin ^22x)\cos 2x}{2}=-\frac{(1+\cos ^22x)\cos 2x}{2}\) (1)
\(-(\frac{7}{8}\cos 2x+\frac{1}{8}\cos 6x)=\frac{-7}{8}\cos 2x-\frac{1}{8}(4\cos ^32x-3\cos 2x)=-\frac{\cos 2x+\cos ^32x}{2}\)
\(=\frac{-\cos 2x(\cos ^22x+1)}{2}\) (2)
Từ $(1);(2)$ ta có đpcm.
1) \(\dfrac{1-cosx+cos2x}{sin2x-sinx}=cotx\)
\(VT=\dfrac{1-cosx+2cos^2x-1}{2sinx.cosx-sinx}\)
\(VT=\dfrac{cosx\left(2cos-1\right)}{sinx\left(2cosx-1\right)}\)
\(VT=\dfrac{cosx}{sinx}=cotx=VP\) ( đpcm )
b) \(\dfrac{sinx+sin\dfrac{x}{2}}{1+cosx+cos\dfrac{x}{2}}=tan\dfrac{x}{2}\)
\(VT=\dfrac{sin\left(2.\dfrac{x}{2}\right)+sin\dfrac{x}{2}}{1+cos\left(2.\dfrac{x}{2}\right)+cos\dfrac{x}{2}}\)
\(VT=\dfrac{2sin\dfrac{x}{2}.cos\dfrac{x}{2}+sin\dfrac{x}{2}}{1+2cos^2\dfrac{x}{2}-1+cos\dfrac{x}{2}}\)
\(VT=\dfrac{2sin\dfrac{x}{2}.cos\dfrac{x}{2}+sin\dfrac{x}{2}}{2cos^2\dfrac{x}{2}+cos\dfrac{x}{2}}\)
\(VT=\dfrac{sin\dfrac{x}{2}\left(2cos\dfrac{x}{2}+1\right)}{cos\dfrac{x}{2}\left(2cos\dfrac{x}{2}+1\right)}\)
\(VT=\dfrac{sin\dfrac{x}{2}}{cos\dfrac{x}{2}}=tan\dfrac{x}{2}=VP\) ( đpcm )
c) \(\dfrac{2cos2x-sin4x}{2cos2x+sin4x}=tan^2\left(\dfrac{\pi}{4}-x\right)\)
\(VT=\dfrac{2cos2x-sin\left(2.2x\right)}{2cos2x+sin\left(2.2x\right)}\)
\(VT=\dfrac{2cos2x-2sin2x.cos2x}{2cos2x+2sin2x.cos2x}\)
\(VT=\dfrac{2cos2x\left(1-sin2x\right)}{2cos2x\left(1+sin2x\right)}\)
\(VT=\dfrac{1-sin2x}{1+sin2x}\)
\(VP=tan^2\left(\dfrac{\pi}{4}-x\right)=\dfrac{1-cos2\left(\dfrac{\pi}{4}-x\right)}{1+cos2\left(\dfrac{\pi}{4}-x\right)}\)
\(VP=\dfrac{1-cos\left(\dfrac{\pi}{2}-2x\right)}{1+cos\left(\dfrac{\pi}{2}-2x\right)}\)
\(VP=\dfrac{1-sin2x}{1+cos2x}=VT\) ( đpcm )
d) \(tanx-tany=\dfrac{sin\left(x-y\right)}{cosx.cosy}\)
\(VP=\dfrac{sin\left(x-y\right)}{cosx.cosy}=\dfrac{sinx.cosy-cosx.siny}{cosx.cosy}\)
\(VP=\dfrac{sinx.cosy}{cosx.cosy}-\dfrac{cosx.siny}{cosx.cosy}\)
\(VP=\dfrac{sinx}{cosx}-\dfrac{siny}{cosy}=tanx-tany=VT\) ( đpcm )
\(A=\frac{2tan15^0}{1-tan^215^0}=tan\left(2.15^0\right)=tan30^0=\frac{\sqrt{3}}{3}\)
\(B=\frac{1}{2}.2sin\frac{\pi}{16}.cos\frac{\pi}{16}.cos\frac{\pi}{8}=\frac{1}{2}.sin\left(2.\frac{\pi}{16}\right)cos\frac{\pi}{8}\)
\(=\frac{1}{4}.2sin\frac{\pi}{8}cos\frac{\pi}{8}=\frac{1}{4}sin\left(2.\frac{\pi}{8}\right)=\frac{1}{4}sin\frac{\pi}{4}=\frac{\sqrt{2}}{8}\)
a) \(\dfrac{tan\alpha-tan\beta}{cot\beta-cot\alpha}=\dfrac{\dfrac{sin\alpha}{cos\alpha}-\dfrac{sin\beta}{cos\beta}}{\dfrac{cos\beta}{sin\beta}-\dfrac{cos\alpha}{sin\alpha}}\)
\(=\dfrac{\dfrac{sin\alpha cos\beta-cos\alpha sin\beta}{cos\alpha cos\beta}}{\dfrac{cos\beta sin\alpha-cos\alpha sin\beta}{sin\beta sin\alpha}}\)
\(=\dfrac{sin\beta sin\alpha}{cos\beta cos\alpha}=tan\alpha tan\beta\).
b) \(tan100^o+\dfrac{sin530^o}{1+sin640^o}=tan100^o+\dfrac{sin170^o}{1+sin280^o}\)
\(=-cot10^o+\dfrac{sin10^o}{1-sin80^o}\)\(=\dfrac{-cos10^o}{sin10^o}+\dfrac{sin10^o}{1-cos10^o}\)
\(=\dfrac{-cos10^o+cos^210^o+sin^210^o}{sin10^o\left(1-cos10^o\right)}\) \(=\dfrac{1-cos10^o}{sin10^o\left(1-cos10^o\right)}=\dfrac{1}{sin10^o}\) .
VT = \(\dfrac{sin}{1+cos}+\dfrac{cos}{sin}\)
= \(\dfrac{sin^2+cos^2+cos}{\left(1+cos\right)sin}\)
= \(\dfrac{1+cos}{\left(1+cos\right)sin}\)
= \(\dfrac{1}{sin}\) = VP (đpcm)