Докажите тождества (их 5)

Question 1

Question 2

1.

$\frac{tg(a)^{2}}{1+tg(a)^{2}}*\frac{1+(\frac{1}{tg(a)})^{2}}{(\frac{1}{tg(a)})^{2}}$ = $\frac{tg(a)^{2}}{1+tg(a)^{2}}*\frac{1+\frac{1}{tg(a)^{2}}}{\frac{1}{tg(a)^{2}}}$ = $\frac{tg(a)^{2}}{1+tg(a)^{2}}*\frac{\frac{tg(a)^{2}+1}{tg(a)^{2}}}{\frac{1}{tg(a)^{2}}}$ = $\frac{tg(a)^{2}}{1+tg(a)^{2}}*\frac{tg(a)^{2}+1}{1}$ = $tg(a)^{2}*\frac{1}{1}$ =tg(a)²×1=tg(a)²;

2.

$(\frac{sin(a)}{cos(a)})^{2}+sin(a)^{2}-\frac{1}{cos(a)^{2}}$ = $\frac{sin(a)^{2}}{cos(a)^{2}}+sin(a)^{2}-\frac{1}{cos(a)^{2}}$ = $\frac{sin(a)^{2}+cos(a)^{2}sin(a)^{2}-1}{cos(a)^{2}}$ = $\frac{-(1-sin(a)^{2})+cos(a)^{2}*sin(a)^{2}}{cos(a)^{2}}$ = $\frac{-cos(a)^{2}+cos(a)^{2}*sin(a)^{2}}{cos(a)^{2}}$ = $\frac{(-1+sin(a)^{2})*cos(a)^{2}}{cos(a)^{2}}$ = $\frac{-(1-sin(a)^{2})cos(a)^{2}}{cos(a)^{2}}$ = $\frac{-cos(a)^{2}*cos(a)^{2}}{cos(a)^{2}}$ = $\frac{-cos(a)^{4}}{cos(a)^{2}}$ =-cos(a)²;

3.

1+ $((\frac{cos(a)}{sin(a)})^{2}-(\frac{sin(a)^{2}}{cos(a)^{2}})^{2})*cos(a)^{2}$ = $1+(\frac{cos(a)^{2}}{sin(a)^{2}}-\frac{sin(a)^{2}}{cos(a)^{2}})*cos(a)^{2}$ = $1+\frac{cos(a)^{4}-sin(a)^{4}}{sin(a)^{2}*cos(a)^{2}}*cos(a)^{2}$ = $1+\frac{-(sin(a)^{4}-cos(a)^{4})}{sin(a)^{2}}$ = $1+\frac{-((sin(a)^{2}-cos(a)^{2})*(sin(a)^{2}+cos(a)^{2}))}{sin(a)^{2}}$ = $1+\frac{-(-(sin(a)^{2}-cos(a)^{2})*1)}{sin(a)^{2}}$ = $1+\frac{-(-cos(2a))}{sin(a)^{2}}$ = $1+\frac{cos(2a)}{sin(a)^{2}}$ = $\frac{sin(a)^{2}+cos(2a)}{sin(a)^{2}}$ = $\frac{sin(a)^{2}+cos(a)^{2}-sin(a)^{2}}{sin(a)^{2}}$ = $\frac{cos(a)^{2}}{sin(a)^{2}}$ = $(\frac{cos(a)}{sin(a)})^{2}$ =ctg(a)²;

4.

cos(2a)+ $\frac{2sin(2a)}{\frac{1}{tg(a)}-tg(a)}$ = $cos(2a)+\frac{2sin(2a)}{\frac{1-tg(a)^{2}}{tg(a)}}$ = $cos(2a)+\frac{2sin(2a)}{\frac{1}{\frac{1}{2}*tg(2a)}}$ = $cos(2a)+\frac{2sin(2a)}{\frac{1}{\frac{tg(2a)}{2}}}$ = $cos(2a)+\frac{2sin(2a)}{\frac{2}{tg(2a)}}$ =cos(2a)+sin(2a)*tg(2a)= $cos(2a)+sin(2a)*\frac{sin(2a)}{cos(2a)}$ =cos(2a)+ $\frac{sin(2a)^{2}}{cos(2a)}$ = $\frac{cos(2a)^{2}+sin(2a)^{2}}{cos(2a)}$ = $\frac{1}{cos(2a)}$ ;

5.

(sin(a)²+(tg(a)·sin(a))²)· $\frac{cos(a)}{sin(a)}$ = $(sin(a)^{2}+(\frac{sin(a)}{cos(a)}*sin(a))^{2})*\frac{cos(a)}{sin(a)}$ =(sin(a)^{2}+<img src="https://tex.z-dn.net/?f=+%28%5Cfrac%7Bsin%28a%29%5E%7B2%7D%7D%7Bcos%28a%29%7D%29%5E%7B2%7D%29%2A%5Cfrac%7Bcos%28a%29%7D%7Bsin%28a%29%7D+" id="TexFormula38" title=" (\frac{sin(a)^{2}}{cos(a)})^{2})*\frac{cos(a)}{sin(a)} " alt=" (\frac{sin(a)^{2}}{cos(a)})^{2})*\frac{cos(a)}{sin(a)} " align="absmi

Pavel2018forever_zn · Answer 1 · 2018-09-10T02:33:57+0000

1.

$\frac{tg(a)^{2}}{1+tg(a)^{2}}*\frac{1+(\frac{1}{tg(a)})^{2}}{(\frac{1}{tg(a)})^{2}}$ = $\frac{tg(a)^{2}}{1+tg(a)^{2}}*\frac{1+\frac{1}{tg(a)^{2}}}{\frac{1}{tg(a)^{2}}}$ = $\frac{tg(a)^{2}}{1+tg(a)^{2}}*\frac{\frac{tg(a)^{2}+1}{tg(a)^{2}}}{\frac{1}{tg(a)^{2}}}$ = $\frac{tg(a)^{2}}{1+tg(a)^{2}}*\frac{tg(a)^{2}+1}{1}$ = $tg(a)^{2}*\frac{1}{1}$ =tg(a)²×1=tg(a)²;

2.

$(\frac{sin(a)}{cos(a)})^{2}+sin(a)^{2}-\frac{1}{cos(a)^{2}}$ = $\frac{sin(a)^{2}}{cos(a)^{2}}+sin(a)^{2}-\frac{1}{cos(a)^{2}}$ = $\frac{sin(a)^{2}+cos(a)^{2}sin(a)^{2}-1}{cos(a)^{2}}$ = $\frac{-(1-sin(a)^{2})+cos(a)^{2}*sin(a)^{2}}{cos(a)^{2}}$ = $\frac{-cos(a)^{2}+cos(a)^{2}*sin(a)^{2}}{cos(a)^{2}}$ = $\frac{(-1+sin(a)^{2})*cos(a)^{2}}{cos(a)^{2}}$ = $\frac{-(1-sin(a)^{2})cos(a)^{2}}{cos(a)^{2}}$ = $\frac{-cos(a)^{2}*cos(a)^{2}}{cos(a)^{2}}$ = $\frac{-cos(a)^{4}}{cos(a)^{2}}$ =-cos(a)²;

3.

1+ $((\frac{cos(a)}{sin(a)})^{2}-(\frac{sin(a)^{2}}{cos(a)^{2}})^{2})*cos(a)^{2}$ = $1+(\frac{cos(a)^{2}}{sin(a)^{2}}-\frac{sin(a)^{2}}{cos(a)^{2}})*cos(a)^{2}$ = $1+\frac{cos(a)^{4}-sin(a)^{4}}{sin(a)^{2}*cos(a)^{2}}*cos(a)^{2}$ = $1+\frac{-(sin(a)^{4}-cos(a)^{4})}{sin(a)^{2}}$ = $1+\frac{-((sin(a)^{2}-cos(a)^{2})*(sin(a)^{2}+cos(a)^{2}))}{sin(a)^{2}}$ = $1+\frac{-(-(sin(a)^{2}-cos(a)^{2})*1)}{sin(a)^{2}}$ = $1+\frac{-(-cos(2a))}{sin(a)^{2}}$ = $1+\frac{cos(2a)}{sin(a)^{2}}$ = $\frac{sin(a)^{2}+cos(2a)}{sin(a)^{2}}$ = $\frac{sin(a)^{2}+cos(a)^{2}-sin(a)^{2}}{sin(a)^{2}}$ = $\frac{cos(a)^{2}}{sin(a)^{2}}$ = $(\frac{cos(a)}{sin(a)})^{2}$ =ctg(a)²;

4.

cos(2a)+ $\frac{2sin(2a)}{\frac{1}{tg(a)}-tg(a)}$ = $cos(2a)+\frac{2sin(2a)}{\frac{1-tg(a)^{2}}{tg(a)}}$ = $cos(2a)+\frac{2sin(2a)}{\frac{1}{\frac{1}{2}*tg(2a)}}$ = $cos(2a)+\frac{2sin(2a)}{\frac{1}{\frac{tg(2a)}{2}}}$ = $cos(2a)+\frac{2sin(2a)}{\frac{2}{tg(2a)}}$ =cos(2a)+sin(2a)*tg(2a)= $cos(2a)+sin(2a)*\frac{sin(2a)}{cos(2a)}$ =cos(2a)+ $\frac{sin(2a)^{2}}{cos(2a)}$ = $\frac{cos(2a)^{2}+sin(2a)^{2}}{cos(2a)}$ = $\frac{1}{cos(2a)}$ ;

5.

(sin(a)²+(tg(a)·sin(a))²)· $\frac{cos(a)}{sin(a)}$ = $(sin(a)^{2}+(\frac{sin(a)}{cos(a)}*sin(a))^{2})*\frac{cos(a)}{sin(a)}$ =(sin(a)^{2}+<img src="https://tex.z-dn.net/?f=+%28%5Cfrac%7Bsin%28a%29%5E%7B2%7D%7D%7Bcos%28a%29%7D%29%5E%7B2%7D%29%2A%5Cfrac%7Bcos%28a%29%7D%7Bsin%28a%29%7D+" id="TexFormula38" title=" (\frac{sin(a)^{2}}{cos(a)})^{2})*\frac{cos(a)}{sin(a)} " alt=" (\frac{sin(a)^{2}}{cos(a)})^{2})*\frac{cos(a)}{sin(a)} " align="absmi