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a) \(tan^2\alpha+cot^2\alpha=\left(tan\alpha+cot\alpha\right)^2-2tan\alpha cot\alpha\)
\(=m^2-2\).
b) \(tan^3\alpha+cot^3\alpha=\left(tan\alpha+cot\alpha\right)\)\(\left(tan^2\alpha-tan\alpha cot\alpha+cot^2\alpha\right)\)
\(=m\left(tan^2\alpha+cot^2\alpha-tan\alpha cot\alpha\right)\)
\(=m\left(m^2-2-2\right)=m\left(m^2-3\right)\).
a) \(tan3\alpha-tan2\alpha-tan\alpha=\left(tan3\alpha-tan\alpha\right)-tan2\alpha\)
\(=\left(\dfrac{sin3\alpha}{cos3\alpha}-\dfrac{sin\alpha}{cos\alpha}\right)-\dfrac{sin2\alpha}{cos2\alpha}\)\(=\dfrac{sin3\alpha cos\alpha-cos3\alpha sin\alpha}{cos3\alpha cos\alpha}-\dfrac{sin2\alpha}{cos2\alpha}\)
\(=\dfrac{sin2\alpha}{cos3\alpha cos\alpha}-\dfrac{sin2\alpha}{cos2\alpha}\)
\(=sin2\alpha.\left(\dfrac{1}{cos3\alpha cos\alpha}-\dfrac{1}{cos2\alpha}\right)\)
\(=sin2\alpha.\dfrac{cos2\alpha-cos3\alpha cos\alpha}{cos3\alpha cos\alpha cos2\alpha}\)
\(=sin2\alpha.\dfrac{cos2\alpha-\dfrac{1}{2}\left(cos4\alpha+cos2\alpha\right)}{cos3\alpha cos2\alpha cos\alpha}\)
\(=sin2\alpha.\dfrac{cos2\alpha-cos4\alpha}{2cos3\alpha cos2\alpha cos\alpha}\)
\(=\dfrac{sin2\alpha.2sin3\alpha.sin\alpha}{2cos3\alpha cos2\alpha cos\alpha}\)
\(=tan3\alpha tan2\alpha tan\alpha\) (Đpcm).
b) \(\dfrac{4tan\alpha\left(1-tan^2\alpha\right)}{\left(1+tan^2\right)^2}=4tan\alpha\left(1-tan^2\alpha\right):\left(\dfrac{1}{cos^2\alpha}\right)^2\)
\(=4tan\alpha\left(1-tan^2\alpha\right)cos^4\alpha\)
\(=4\dfrac{sin\alpha}{cos\alpha}\left(1-\dfrac{sin^2\alpha}{cos^2\alpha}\right)cos^4\alpha\)
\(=4sin\alpha\left(cos^2\alpha-sin^2\alpha\right)cos\alpha\)
\(=4sin\alpha cos\alpha.cos2\alpha\)
\(=2.sin2\alpha.cos2\alpha=sin4\alpha\) (Đpcm).
a) \(sin6\alpha cot3\alpha cos6\alpha=2.sin3\alpha.cos3\alpha\dfrac{cos3\alpha}{sin3\alpha}-cos6\alpha\)
\(=2cos^23\alpha-\left(2cos^23\alpha-1\right)=1\) (Không phụ thuộc vào x).
b) \(\left[tan\left(90^o-\alpha\right)-cot\left(90^o+\alpha\right)\right]^2\)\(-\left[cot\left(180^o+\alpha\right)+cot\left(270^o+\alpha\right)\right]^2\)
\(=\left[cot\alpha+cot\left(90^o-\alpha\right)\right]^2\)\(-\left[cot\alpha+cot\left(90^o+\alpha\right)\right]^2\)
\(=\left[cot\alpha+tan\alpha\right]^2-\left[cot\alpha-tan\alpha\right]^2\)
\(=4tan\alpha cot\alpha=4\). (Không phụ thuộc vào \(\alpha\)).
\(A+B+C=180^0\Rightarrow\frac{A}{2}+\frac{B}{2}+\frac{C}{2}=90^0\Rightarrow\frac{A}{2}+\frac{B}{2}=90^0-\frac{C}{2}\)
\(\Rightarrow tan\left(\frac{A}{2}+\frac{B}{2}\right)=tan\left(90^0-\frac{C}{2}\right)\)
\(\Leftrightarrow\frac{tan\frac{A}{2}+tan\frac{B}{2}}{1-tan\frac{A}{2}.tan\frac{B}{2}}=cot\frac{C}{2}=\frac{1}{tan\frac{C}{2}}\)
\(\Leftrightarrow tan\frac{C}{2}\left(tan\frac{A}{2}+tan\frac{B}{2}\right)=1-tan\frac{A}{2}.tan\frac{B}{2}\)
\(\Leftrightarrow tan\frac{A}{2}tan\frac{C}{2}+tan\frac{B}{2}tan\frac{C}{2}+tan\frac{A}{2}.tan\frac{B}{2}=1\)
b/\(A+B+C=180^0\Rightarrow A+B=180^0-C\)
\(\Rightarrow cot\left(A+B\right)=cot\left(180^0-C\right)\)
\(\Leftrightarrow\frac{cotA.cotB-1}{cotA+cotB}=-cotC\)
\(\Leftrightarrow cotA.cotB-1=-cotA.cotC-cotB.cotC\)
\(\Leftrightarrow cotA.cotB+cotB.cotC+cotA.cotC=1\)
\(A=tan\left(a+b\right)=tan\frac{\pi}{4}=1\)
Ta có: \(tan\left(a+b\right)=\frac{tana+tanb}{1-tana.tanb}\)
\(\Rightarrow B=tana+tanb=tan\left(a+b\right)\left(1-tana.tanb\right)=1.\left(1-3+2\sqrt{2}\right)=2\sqrt{2}-2\)
\(\left\{{}\begin{matrix}tana+tanb=2\sqrt{2}-2\\tana.tanb=3-2\sqrt{2}\end{matrix}\right.\)
Theo Viet đảo, \(tana;tanb\) là nghiệm của:
\(x^2-\left(2\sqrt{2}-2\right)x+3-2\sqrt{2}=0\)
\(\Leftrightarrow\left(x-\sqrt{2}+1\right)^2=0\Rightarrow x=\sqrt{2}-1\)
\(\Rightarrow tana=tanb=\sqrt{2}-1\Rightarrow a=b=\frac{\pi}{8}\)
\(tan^2a+cot^2a=\left(tana+cota\right)^2-2=m^2-2\)
\(tan^4a+cot^4a=\left(tan^2a+cot^2a\right)^2-2=\left(m^2-2\right)^2-2\)
\(tan^6a+cot^6a=\left(tan^2a+cot^2a\right)^3-3\left(tan^2a+cot^2a\right)\)
\(=\left(m^2-2\right)^3-3\left(m^2-2\right)\)
\(m^2=\left(tana+cota\right)^2=\left(tana-cota\right)^2+4tana.cota\)
\(\Rightarrow m^2=\left(tana-cota\right)^2+4\ge4\)
\(\Rightarrow\left|m\right|\ge2\)
Do \(\dfrac{\pi}{2}< \alpha< \pi\) nên \(tan\alpha< 0,cot\alpha< 0;cos\alpha< 0\).
Vì vậy: \(cos\alpha=-\sqrt{1-sin^2\alpha}=-\dfrac{\sqrt{7}}{4}\).
\(tan\alpha=\dfrac{sin\alpha}{cos\alpha}=\dfrac{3}{4}:\dfrac{-\sqrt{7}}{4}=\dfrac{-3}{\sqrt{7}}\).
\(cot\alpha=\dfrac{1}{tan\alpha}=\dfrac{-\sqrt{7}}{3}\).
\(A=\dfrac{2tan\alpha-3cot\alpha}{cos\alpha+tan\alpha}\)\(=\dfrac{2.\dfrac{-3}{\sqrt{7}}-3.\dfrac{-\sqrt{7}}{3}}{\dfrac{-\sqrt{7}}{4}+\dfrac{-3}{\sqrt{7}}}\)
\(=\dfrac{\dfrac{-6}{\sqrt{7}}+\sqrt{7}}{\dfrac{-7-12}{4\sqrt{7}}}\)\(=\dfrac{\dfrac{-6+7}{\sqrt{7}}.4\sqrt{7}}{-19}\)\(=\dfrac{\dfrac{1}{\sqrt{7}}.4\sqrt{7}}{-19}=-\dfrac{4}{19}\).
b) \(\dfrac{cos^2\alpha+cot^2\alpha}{tan\alpha-cot\alpha}=\dfrac{\left(-\dfrac{\sqrt{7}}{4}\right)^2+\left(\dfrac{-\sqrt{7}}{3}\right)^2}{\dfrac{-3}{\sqrt{7}}+\dfrac{\sqrt{7}}{3}}\)
\(=\dfrac{\dfrac{7}{16}+\dfrac{7}{9}}{\dfrac{-9+7}{3\sqrt{7}}}=\dfrac{\dfrac{175}{144}}{\dfrac{-2}{3\sqrt{7}}}=\dfrac{-175}{96\sqrt{7}}\).