Доказать тождество: ctgx/ctgx+tgx=cos^2x

$\frac{Ctgx}{Ctgx+tgx}= \frac{Ctgx}{Ctgx+ \frac{1}{Ctgx} } = \frac{Ctg ^{2}x }{1+Ctg ^{2}x }= \frac{ \frac{Cos ^{2}x }{Sin ^{2} x} }{1+ \frac{Cos ^{2} x}{Sin ^{2} x} } = \frac{ \frac{Cos ^{2} x}{Sin ^{2} x} }{ \frac{Sin ^{2} x+Cos ^{2} x}{Sin ^{2} x} }= \frac{ \frac{Cos ^{2} x}{Sin ^{2} x} }{ \frac{1}{Sin ^{2} x} }=$ $\frac{Cos ^{2} x*Sin ^{2} x}{Sin ^{2} x}=Cos ^{2}x\\\\Cos ^{2} x=Cos ^{2} x$
тождество доказано