Question

докажите тождества (их 5)

Answer 1

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