Question

докажите тождество....​

Answer 1

$\Big((cos2a)^{-1}+ctg\Big(\dfrac{5\pi}{2}+2a\Big)\Big)\cdot ctg\Big(\dfrac{5\pi }{4}-a\Big)=\\\\\\=\Big(\dfrac{1}{cos2a}+ctg\Big(2\pi +\dfrac{\pi}{2}+2a\Big)\Big)\cdot ctg\Big(\pi +\dfrac{\pi}{4}-a\Big)=\\\\\\=\Big(\dfrac{1}{cos2a}+ctg\Big(\dfrac{\pi}{2}+2a\Big)\Big)\cdot ctg\Big(\dfrac{\pi}{4}-a\Big)=\\\\\\=\Big(\dfrac{1}{cos2a}-tg(2a)\Big)\cdot ctg\Big(\dfrac{\pi}{4}-a\Big)=\dfrac{1-sin2a}{cos2a}\cdot \dfrac{1+tg\frac{\pi}{4}\cdot tga}{tg\frac{\pi}{4}-tga}=$

$=\dfrac{sin^2a+cos^2a-2sina\cdot cosa}{cos^2a-sin^2a}\cdot \dfrac{1+\frac{sina}{cosa}}{1-\frac{sina}{cosa}}=\\\\\\=\dfrac{(cosa-sina)^2}{(cosa-sina)(cosa+sina)}\cdot \dfrac{cosa+sina}{cosa-sina}=\dfrac{cosa-sina}{cosa+sina}\cdot \dfrac{cosa+sina}{cosa-sina}=1\\\\\\1=1$