\(VT=\frac{2\cos2\alpha.\cos\alpha}{2.\sin2\alpha\cos\alpha}.\frac{\sin2\alpha}{\cos2\alpha}-2\left(2\sin\alpha.\cos\alpha\right)^2\)

\(VT=1-2\left(\sin2\alpha\right)^2=\cos4\alpha\)

\(\frac{1-cosa+cos2a}{sin2a-sina}=\frac{1-cosa+2cos^2a-1}{2sina.cosa-sina}=\frac{cosa\left(2cosa-1\right)}{sina\left(2cosa-1\right)}=\frac{cosa}{sina}=cota\)

\(a,\dfrac{1}{tan\alpha+1}+\dfrac{1}{cot\alpha+1}\\ =\dfrac{cot\alpha+1+tan\alpha+1}{\left(tan\alpha+1\right)\left(cot\alpha+1\right)}\\ =\dfrac{tan\alpha+cot\alpha+2}{tan\alpha\cdot cot\alpha+tan\alpha+cot\alpha+1}\\ =\dfrac{tan\alpha+cot\alpha+2}{tan\alpha+cot\alpha+2}\\ =1\)

\(b,cos\left(\dfrac{\pi}{2}-\alpha\right)-sin\left(\pi+\alpha\right)\\ =sin\alpha+sin\alpha\\ =2sin\alpha\)

\(c,sin\left(\alpha-\dfrac{\pi}{2}\right)+cos\left(-\alpha+6\pi\right)-tan\left(\alpha+\pi\right)cot\left(3\pi-\alpha\right)\\ =-sin\left(\dfrac{\pi}{2}-\alpha\right)+cos\left(\alpha\right)-tan\left(\alpha\right)cot\left(\pi-\alpha\right)\\ =-cos\left(\alpha\right)+cos\left(\alpha\right)+tan\left(\alpha\right)\cdot cot\left(\alpha\right)\\ =1\)

a) \(\frac{1-\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1+\cos a}\)

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

\(\Leftrightarrow1-\cos^2\alpha=\sin^2\alpha\)

\(\Leftrightarrow\sin^2\alpha+\cos^2\alpha=1\)( luôn đúng )

\(\Rightarrow\frac{1-\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1+\cos\alpha}\)

Tham khảo:

a) 

Gọi M(x;y) là điểm trên đường tròn đơn vị sao cho \(\widehat {xOM} = \alpha \). Gọi N, P tương ứng là hình chiếu vuông góc của M lên các trục Ox, Oy.

Ta có: \(\left\{ \begin{array}{l}x = \cos \alpha \\y = \sin \alpha \end{array} \right. \Rightarrow \left\{ \begin{array}{l}{\cos ^2}\alpha  = {x^2}\\{\sin ^2}\alpha  = {y^2}\end{array} \right.\)(1)

Mà \(\left\{ \begin{array}{l}\left| x \right| = ON\\\left| y \right| = OP = MN\end{array} \right. \Rightarrow \left\{ \begin{array}{l}{x^2} = {\left| x \right|^2} = O{N^2}\\{y^2} = {\left| y \right|^2} = M{N^2}\end{array} \right.\)(2)

Từ (1) và (2) suy ra \({\sin ^2}\alpha  + {\cos ^2}\alpha  = O{N^2} + M{N^2} = O{M^2}\) (do \(\Delta OMN\) vuông tại N)

\( \Rightarrow {\sin ^2}\alpha  + {\cos ^2}\alpha  = 1\) (vì OM =1). (đpcm)

b) 

Ta có:  \(\tan \alpha  = \frac{{\sin \alpha }}{{\cos \alpha }}\;\;(\alpha  \ne {90^o})\)

\( \Rightarrow 1 + {\tan ^2}\alpha  = 1 + \frac{{{{\sin }^2}\alpha }}{{{{\cos }^2}\alpha }} = \frac{{{{\cos }^2}\alpha }}{{{{\cos }^2}\alpha }} + \frac{{{{\sin }^2}\alpha }}{{{{\cos }^2}\alpha }} = \frac{{{{\sin }^2}\alpha  + {{\cos }^2}\alpha }}{{{{\cos }^2}\alpha }}\)

Mà theo ý a) ta có \({\sin ^2}\alpha  + {\cos ^2}\alpha  = 1\) với mọi góc \(\alpha \)

\( \Rightarrow 1 + {\tan ^2}\alpha  = \frac{1}{{{{\cos }^2}\alpha }}\) (đpcm)

c) 

Ta có:  \(\cot \alpha  = \frac{{\cos \alpha }}{{\sin \alpha }}\;\;\;({0^o} < \alpha  < {180^o})\)

\( \Rightarrow 1 + {\cot ^2}\alpha  = 1 + \frac{{{{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }} = \frac{{{{\sin }^2}\alpha }}{{{{\sin }^2}\alpha }} + \frac{{{{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }} = \frac{{{{\sin }^2}\alpha  + {{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }}\)

Mà theo ý a) ta có \({\sin ^2}\alpha  + {\cos ^2}\alpha  = 1\) với mọi góc \(\alpha \)

\( \Rightarrow 1 + {\cot ^2}\alpha  = \frac{1}{{{{\sin }^2}\alpha }}\) (đpcm)

Help help. Tui thật sự ngu lượng giác huhu

a) Vì \(0<\alpha <\frac{\pi }{2} \) nên \(\sin \alpha  > 0\). Mặt khác, từ \({\sin ^2}\alpha  + {\cos ^2}\alpha  = 1\) suy ra

\(\sin \alpha  = \sqrt {1 - {{\cos }^2}a}  = \sqrt {1 - \frac{1}{{25}}}  = \frac{{2\sqrt 6 }}{5}\)

Do đó, \(\tan \alpha  = \frac{{\sin \alpha }}{{\cos \alpha }} = \frac{{\frac{{2\sqrt 6 }}{5}}}{{\frac{1}{5}}} = 2\sqrt 6 \) và \(\cot \alpha  = \frac{{\cos \alpha }}{{\sin \alpha }} = \frac{{\frac{1}{5}}}{{\frac{{2\sqrt 6 }}{5}}} = \frac{{\sqrt 6 }}{{12}}\)

b) Vì \(\frac{\pi }{2} < \alpha  < \pi\) nên \(\cos \alpha  < 0\). Mặt khác, từ \({\sin ^2}\alpha  + {\cos ^2}\alpha  = 1\) suy ra

       \(\cos \alpha  = \sqrt {1 - {{\sin }^2}a}  = \sqrt {1 - \frac{4}{9}}  = -\frac{{\sqrt 5 }}{3}\)

Do đó, \(\tan \alpha  = \frac{{\sin \alpha }}{{\cos \alpha }} = \frac{{\frac{2}{3}}}{{-\frac{{\sqrt 5 }}{3}}} = -\frac{{2\sqrt 5 }}{5}\) và \(\cot \alpha  = \frac{{\cos \alpha }}{{\sin \alpha }} = \frac{{-\frac{{\sqrt 5 }}{3}}}{{\frac{2}{3}}} = -\frac{{\sqrt 5 }}{2}\)

c) Ta có: \(\cot \alpha  = \frac{1}{{\tan \alpha }} = \frac{1}{{\sqrt 5 }}\)

Ta có: \({\tan ^2}\alpha  + 1 = \frac{1}{{{{\cos }^2}\alpha }} \Rightarrow {\cos ^2}\alpha  = \frac{1}{{{{\tan }^2}\alpha  + 1}} = \frac{1}{6} \Rightarrow \cos \alpha  =  \pm \frac{1}{{\sqrt 6 }}\)

Vì \(\pi  < \alpha  < \frac{{3\pi }}{2} \Rightarrow \sin \alpha  < 0\;\) và \(\,\,\cos \alpha  < 0 \Rightarrow \cos \alpha  = -\frac{1}{{\sqrt 6 }}\)

Ta có: \(\tan \alpha  = \frac{{\sin \alpha }}{{\cos \alpha }} \Rightarrow \sin \alpha  = \tan \alpha .\cos \alpha  = \sqrt 5 .(-\frac{1}{{\sqrt 6 }}) = -\sqrt {\frac{5}{6}} \)

d) Vì \(\cot \alpha  =  - \frac{1}{{\sqrt 2 }}\;\,\) nên \(\,\,\tan \alpha  = \frac{1}{{\cot \alpha }} =  - \sqrt 2 \)

Ta có: \({\cot ^2}\alpha  + 1 = \frac{1}{{{{\sin }^2}\alpha }} \Rightarrow {\sin ^2}\alpha  = \frac{1}{{{{\cot }^2}\alpha  + 1}} = \frac{2}{3} \Rightarrow \sin \alpha  =  \pm \sqrt {\frac{2}{3}} \)

Vì \(\frac{{3\pi }}{2} < \alpha  < 2\pi  \Rightarrow \sin \alpha  < 0 \Rightarrow \sin \alpha  =  - \sqrt {\frac{2}{3}} \)

Ta có: \(\cot \alpha  = \frac{{\cos \alpha }}{{\sin \alpha }} \Rightarrow \cos \alpha  = \cot \alpha .\sin \alpha  = \left( { - \frac{1}{{\sqrt 2 }}} \right).\left( { - \sqrt {\frac{2}{3}} } \right) = \frac{{\sqrt 3 }}{3}\)