Question

Надо упростить выражение,молю

Answer 1

$1)\; \; (tg\beta +ctg\beta )^2-(tg\beta -ctg\beta )^2=[\, A^2-B^2=(A-B)(A+B)\, ]=\\\\=(tg\beta +ctg\beta -(tg\beta -ctg\beta ))\cdot (tg\beta +ctg\beta +(tg\beta -ctg\beta ))=\\\\=2ctg\beta \cdot 2tg\beta =4\cdot (\underbrace {tg\beta \cdot ctg\beta }_{1})=4\cdot 1=4$

$2)\; \; tgx+\frac{cosx}{1+sinx} =\frac{sinx}{cosx}+\frac{cosx}{1+sinx}=\frac{sinx(1+sinx)+cos^2x}{cosx(1+sinx)}=\\\\=\frac{sinx+sin^2x+cos^2x}{cosx(1+sinx)}=\frac{sinx+1}{cosx(1+sinx)} =\frac{1}{cosx}\\\\3)\; \; \frac{1-sina}{cosa}-\frac{cosa}{1+sina}=\frac{(1-sina)(1+sina)-cos^2a}{cosa(1+sina)}=\frac{1-sin^2a-cos^2a}{cosa(1+sina)}=\\\\=\frac{1-(sin^2a+cos^2a)}{cosa(1+sina)}=\frac{1-1}{cosa(1+sina)}=0$