Question

ctg(45-a) - ctg(45+a)

Answer 1

$\star \; \; ctgx-ctgy=\frac{cosx}{sinx}-\frac{cosy}{siny}=\frac{cosx\, siny-sinx\, cosy}{sinx\cdot siny}=\frac{sin(y-x)}{sinx\cdot siny}\; \star \\\\\\ctg(45-a)-ctg(45+a)=\frac{sin(45+a-(45-a))}{sin(45-a)\cdot sin(45+a)}=\frac{sin2a}{\frac{1}{2}(cos2a-cos90)} =\\\\= \frac{2\, sin2a}{cos2a} =2\cdot tg2a\\\\\\\star \; \; sinx\cdot siny=\frac{1}{2}\cdot (cos(x-y)-cos(x+y))\; \; \star$