Предмет: Алгебра
Автор: Nishinoyaa
Вопрос задан 8 лет назад

докажите тождества (их 5)

Приложения:

Ответы

Ответ дал: Аноним

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

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

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;

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)+ $frac{sin(2a)^{2}}{cos(2a)}$ = $frac{cos(2a)^{2}+sin(2a)^{2}}{cos(2a)}$ = $frac{1}{cos(2a)}$ ;

(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)