Докажите тождества [latex] \frac{1-tg^{2}(45 -a)}{1+tg^{2}(45-a)}=sin2a [/latex]

[latex] tg( \alpha \pm \beta )= \frac{tg \alpha \pm tg \beta }{1\mp tg \alpha tg \beta } \\\ tg( 45-a )= \frac{tg45-tg a}{1+tg 45tg a} = \frac{1-tg a}{1+tg a} [/latex] [latex] \frac{1-tg^{2}(45 -a)}{1+tg^{2}(45-a)}= \cfrac{1- (\frac{1-tg a}{1+tg a})^2 }{1+(\frac{1-tg a}{1+tg a})^2}= \cfrac{1- \frac{1-2tg a+tg^2a}{1+2tg a+tg^2a} }{1+\frac{1-2tg a+tg^2a}{1+2tg a+tg^2a}}= \\\ =\cfrac{\frac{1+2tg a+tg^2a-1+2tg a-tg^2a}{1+2tg a+tg^2a} }{\frac{1+2tg a+tg^2a+1-2tg a+tg^2a}{1+2tg a+tg^2a}}= \frac{1+2tg a+tg^2a-1+2tg a-tg^2a}{1+2tg a+tg^2a+1-2tg a+tg^2a}= \frac{4tg a}{2+2tg^2a}= \\\ =\frac{2tg a}{1+tg^2a}= \frac{2sina}{cosa} \cdot cos^2a=2 sinacosa=sin2a[/latex]