a) Vì \(\frac{\pi }{2} < a < \pi \) nên \(\cos a < 0\). Do đó \(\cos a = \sqrt {1 - {{\sin }^2}a}  = \sqrt {1 - \frac{1}{3}}  =  - \frac{{\sqrt 6 }}{3}\)

Ta có: \(\cos \left( {a + \frac{\pi }{6}} \right) = \cos a\cos \frac{\pi }{6} - \sin a\sin \frac{\pi }{6} =  - \frac{{\sqrt 6 }}{3}.\frac{{\sqrt 3 }}{2} - \frac{1}{{\sqrt 3 }}.\frac{1}{2} =  - \frac{{\sqrt 3  + 3\sqrt 2 }}{6}\)

b) Vì \(\pi  < a < \frac{{3\pi }}{2}\) nên \(\sin a < 0\). Do đó \(\sin a = \sqrt {1 - {{\cos }^2}a}  = \sqrt {1 - \frac{1}{9}}  =  - \frac{{2\sqrt 2 }}{3}\)

Suy ra \(\tan a\; = \frac{{\sin a}}{{\cos a}} = \frac{{ - \frac{{2\sqrt 2 }}{3}}}{{ - \frac{1}{3}}} = 2\sqrt 2 \)

Ta có: \(\tan \left( {a - \frac{\pi }{4}} \right) = \frac{{\tan a - \tan \frac{\pi }{4}}}{{1 + \tan a\tan \frac{\pi }{4}}} = \frac{{\frac{{\sin a}}{{\cos a}} - 1}}{{1 + \frac{{\sin a}}{{\cos a}}}} = \frac{{2\sqrt 2  - 1}}{{1 + 2\sqrt 2 }} = \frac{{9 - 4\sqrt 2 }}{7}\)

Mn ơi cứu tui

Câu a)

Từ \(\tan a=3\Leftrightarrow \frac{\sin a}{\cos a}=3\Rightarrow \sin a=3\cos a\)

Do đó:

\(\frac{\sin a\cos a+\cos ^2a}{2\sin ^2a-\cos ^2a}=\frac{3\cos a\cos a+\cos ^2a}{2(3\cos a)^2-\cos ^2a}\)

\(=\frac{\cos ^2a(3+1)}{\cos ^2a(18-1)}=\frac{4}{17}\)

Câu b)

Có: \(\cot \left(\frac{\pi}{2}-x\right)=\tan x=\frac{\sin x}{\cos x}\)

\(\cos\left(\frac{\pi}{2}+x\right)=-\sin x\)

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

Và:

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

Do đó:

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

Ta có đpcm.

a) Vì \(\frac{\pi }{2} < a < \pi \) nên \(\cos a < 0\)

Ta có: \({\sin ^2}a + {\cos ^2}a  = 1\)

 \(\Leftrightarrow \frac{1}{9} + {\cos ^2}a  = 1\)

\(\Leftrightarrow {\cos ^2}a =  1 - \frac{1}{9}= \frac{8}{9}\)

\(\Leftrightarrow \cos a  =\pm\sqrt { \frac{8}{9}}  =  \pm \frac{{2\sqrt 2 }}{3}\)

Vì \(\cos a < 0\) nên \(cos a =-\frac{{2\sqrt 2 }}{3}\)

Suy ra \(\tan a = \frac{{\sin a}}{{\cos a}} = \frac{{\frac{1}{3}}}{{ - \frac{{2\sqrt 2 }}{3}}} =  - \frac{{\sqrt 2 }}{4}\)

Ta có: \(\sin 2a = 2\sin a\cos a = 2.\frac{1}{3}.\left( { - \frac{{2\sqrt 2 }}{3}} \right) =  - \frac{{4\sqrt 2 }}{9}\)

\(\cos 2a = 1 - 2{\sin ^2}a = 1 - \frac{2}{9} = \frac{7}{9}\)

\(\tan 2a = \frac{{2\tan a}}{{1 - {{\tan }^2}a}} = \frac{{2.\left( { - \frac{{\sqrt 2 }}{4}} \right)}}{{1 - {{\left( { - \frac{{\sqrt 2 }}{4}} \right)}^2}}} =  - \frac{{4\sqrt 2 }}{7}\)

b) Vì \(\frac{\pi }{2} < a < \frac{{3\pi }}{4}\) nên \(\sin a > 0,\cos a < 0\)

\({\left( {\sin a + \cos a} \right)^2} = {\sin ^2}a + {\cos ^2}a + 2\sin a\cos a = 1 + 2\sin a\cos a = \frac{1}{4}\)

Suy ra \(\sin 2a = 2\sin a\cos a = \frac{1}{4} - 1 =  - \frac{3}{4}\)

Ta có: \({\sin ^2}a + {\cos ^2}a = 1\;\)

\( \Leftrightarrow \left( {\frac{1}{2} - {\cos }a} \right)^2 + {\cos ^2}a - 1 = 0\)

\( \Leftrightarrow \frac{1}{4} - \cos a + {\cos ^2}a + {\cos ^2}a - 1 = 0\)

\( \Leftrightarrow 2{\cos ^2}a - \cos a - \frac{3}{4} = 0\)

\( \Rightarrow \cos a = \frac{{1 - \sqrt 7 }}{4}\) (Vì \(\cos a < 0)\)

\(\cos 2a = 2{\cos ^2}a - 1 = 2.{\left( {\frac{{1 - \sqrt 7 }}{4}} \right)^2} - 1 =  - \frac{{\sqrt 7 }}{4}\)

\(\tan 2a = \frac{{\sin 2a}}{{\cos 2a}} = \frac{{ - \frac{3}{4}}}{{ - \frac{{\sqrt 7 }}{4}}} = \frac{{3\sqrt 7 }}{7}\)

\(E=\frac{cosx}{sinx}+\frac{sinx}{1+cosx}=\frac{cosx+cos^2x+sin^2x}{sinx\left(1+cosx\right)}=\frac{cosx+1}{sinx\left(1+cosx\right)}=\frac{1}{sinx}\)

17.

\(\frac{\pi}{2}< a< \pi\Rightarrow cosa< 0\Rightarrow cosa=-\sqrt{1-sin^2a}=-\frac{12}{13}\)

\(0< b< \frac{\pi}{2}\Rightarrow sinb>0\Rightarrow sinb=\sqrt{1-cos^2b}=\frac{4}{5}\)

\(sin\left(a+b\right)=sina.cosb+cosa.sinb=\frac{5}{13}.\frac{3}{5}-\frac{12}{13}.\frac{4}{5}=-\frac{33}{65}\)

18.

\(K=sin\frac{2\pi}{7}+sin\frac{6\pi}{7}+sin\frac{4\pi}{7}\)

\(\Leftrightarrow K.sin\frac{\pi}{7}=sin\frac{\pi}{7}.sin\frac{2\pi}{7}+sin\frac{\pi}{7}.sin\frac{4\pi}{7}+sin\frac{\pi}{7}.sin\frac{6\pi}{7}\)

\(=\frac{1}{2}\left(cos\frac{\pi}{7}-cos\frac{3\pi}{7}+cos\frac{\pi}{7}-cos\frac{5\pi}{7}+cos\frac{5\pi}{7}-cos\frac{7\pi}{7}\right)\)

\(=\frac{1}{2}\left(cos\frac{\pi}{7}-cos\pi\right)=\frac{1}{2}\left(cos\frac{\pi}{7}+1\right)=\frac{1}{2}\left(2cos^2\frac{\pi}{14}-1+1\right)=cos^2\frac{\pi}{14}\)

\(\Leftrightarrow K.2.sin\frac{\pi}{14}.cos\frac{\pi}{14}=cos^2\frac{\pi}{14}\)

\(\Leftrightarrow2K=\frac{cos\frac{\pi}{14}}{sin\frac{\pi}{14}}=cot\frac{\pi}{14}=a\Rightarrow K=\frac{a}{2}\)

Do \(\pi< \alpha< \dfrac{3\pi}{2}\) nên \(sin\alpha,cos\alpha< 0;tan\alpha,cot\alpha< 0\).
\(cos\left(\alpha-\dfrac{\pi}{2}\right)=cos\left(\dfrac{\pi}{2}-\alpha\right)=sin\alpha< 0\).
\(sin\left(\dfrac{\pi}{2}+\alpha\right)=cos\alpha< 0\).
\(tan\left(\dfrac{3\pi}{2}-\alpha\right)=tan\left(\dfrac{3\pi}{2}-\alpha-2\pi\right)\)\(=tan\left(-\dfrac{\pi}{2}-\alpha\right)\)\(=-tan\left(\dfrac{\pi}{2}+\alpha\right)=cot\left(\alpha\right)>0\).
\(cot\left(\alpha+\pi\right)=cot\left(\alpha\right)>0\).