1: 

a: sin a=căn 3/2

\(cosa=\sqrt{1-sin^2a}=\sqrt{1-\dfrac{3}{4}}=\sqrt{\dfrac{1}{4}}=\dfrac{1}{2}\)

\(tana=\dfrac{\sqrt{3}}{2}:\dfrac{1}{2}=\sqrt{3}\)

cot a=1/tan a=1/căn 3

b: \(tana=2\)

=>cot a=1/tan a=1/2

\(1+tan^2a=\dfrac{1}{cos^2a}\)

=>\(\dfrac{1}{cos^2a}=5\)

=>cos^2a=1/5

=>cosa=1/căn 5

\(sina=\sqrt{1-cos^2a}=\sqrt{\dfrac{4}{5}}=\dfrac{2}{\sqrt{5}}\)

c: \(cosa=\sqrt{1-\left(\dfrac{5}{13}\right)^2}=\dfrac{12}{13}\)

tan a=5/13:12/13=5/12

cot a=1:5/12=12/5

D

D

Cos^2(a) = 1/(1+tan^2(a)) = 4/13

--> cosa = 2sqrt(13)/13

Sin^2(a)=1-4/13=9/13

--> sina = 3sqrt(13)/13

Cota = 2/3 --> tana = 3/2

`sin^2 α+cos^2α=1`

`<=> (2/3)^2+cos^2α=1`

`=> cosα= \sqrt5/3`

`=> tan α=(sinα)/(cosα) = (2\sqrt5)/5`

`=> cota = 1/(tanα)=sqrt5/2`

Vì 0 < α < π/2 nên sin α > 0, cos α > 0, tan α > 0, cot α > 0.

`\pi/2 < \alpha < \pi=>\alpha` nằm ở góc phần tư thứ `2`

    `=>{(sin  \alpha > 0;cos \alpha < 0),(tan \alpha < 0; cot \alpha < 0):}`

      `->\bb D`

2tan a-cot a=1

=>2tana-1/tan a=1

=>\(\dfrac{2tan^2a-1}{tana}=1\)

=>2tan^2a-tana-1=0

=>(tan a-1)(2tana+1)=0

=>tan a=-1/2 hoặc tan a=1

\(P=\dfrac{tan\left(-a\right)+2\cdot cota}{3\cdot tan\left(\dfrac{pi}{2}+a\right)}=\dfrac{-tana+2\cdot cota}{-3\cdot cota}\)

TH1: tan a=-1/2

\(P=\dfrac{\dfrac{1}{2}+2\cdot\left(-2\right)}{-3\cdot\left(-2\right)}=-\dfrac{7}{2}:6=-\dfrac{7}{12}\)

TH2: tan a=1

=>cot a=1

\(P=\dfrac{-1+2}{-3}=\dfrac{1}{-3}=-\dfrac{1}{3}\)

Ta có :

\(2tan\alpha-cot\alpha=1\)

\(\Leftrightarrow2tan\alpha-\dfrac{1}{tan\alpha}=1\)

\(\Leftrightarrow2tan\alpha-\dfrac{1}{tan\alpha}-1=0\)

\(\Leftrightarrow\dfrac{2tan^2\alpha-tan\alpha-1}{tan\alpha}=0\left(tan\alpha\ne0\right)\)

\(\Leftrightarrow2tan^2\alpha-tan\alpha-1=0\)

\(\Leftrightarrow\left[{}\begin{matrix}tan\alpha=1\\tan\alpha=-\dfrac{1}{2}\end{matrix}\right.\)

\(P=\dfrac{tan\left(8\pi-\alpha\right)+2cot\left(\pi+\alpha\right)}{3tan\left(\dfrac{3\pi}{2}+\alpha\right)}\)

\(\Leftrightarrow P=\dfrac{tan\left(4.2\pi-\alpha\right)+2cot\alpha}{3tan\left(2\pi-\dfrac{\pi}{2}+\alpha\right)}\)

\(\Leftrightarrow P=\dfrac{tan\left(-\alpha\right)+2cot\alpha}{3tan\left[-\left(\dfrac{\pi}{2}-\alpha\right)\right]}\)

\(\Leftrightarrow P=\dfrac{-tan\alpha+2cot\alpha}{-3tan\left(\dfrac{\pi}{2}-\alpha\right)}\)

\(\Leftrightarrow P=\dfrac{-tan\alpha+2cot\alpha}{-3cot\alpha}\)

- Với \(tan\alpha=1\Rightarrow cot\alpha=1\)

\(\Leftrightarrow P=\dfrac{-1+2.1}{-3.1}=-\dfrac{1}{3}\)

- Với \(tan\alpha=-\dfrac{1}{2}\Rightarrow cot\alpha=-2\)

\(\Leftrightarrow P=\dfrac{\dfrac{1}{2}+2.\left(-2\right)}{-3.\left(-2\right)}=\dfrac{-\dfrac{7}{2}}{6}=-\dfrac{7}{12}\)

\(tan\left(\dfrac{3\pi}{2}-\alpha\right)+cot\left(3\pi-\alpha\right)-cos\left(\dfrac{\pi}{2}-\alpha\right)+2.sin\left(\pi+\alpha\right)\)

\(=tan\left(\pi+\dfrac{\pi}{2}-\alpha\right)+cot\left(-\alpha\right)-sin\alpha+2\left(sin\pi.cos\alpha+cos\pi.sin\alpha\right)\)

\(=tan\left(\dfrac{\pi}{2}-\alpha\right)-cot\alpha-sin\alpha+2.-sin\alpha\)

\(=cot\alpha-cot\alpha-3sin\alpha\)

\(=-3sin\alpha\)