Доказать тождество:

 

(sin t +cos t)^2 - 1/ (ctg t - sin t*cos t) = 
=(sin t)^2 +2sin t*cos t+(cos t)^2 - ((sin t)^2+(cos t)^2)/ (ctg t - sin t*cos t) = 

=(sin t)^2 +2sin t*cos t+(cos t)^2 - (sin t)^2-(cos t)^2/ (ctg t - sin t*cos t) = 
=2sin t*cos t/ (cos t/sin t) - sin t*cos t
= 2sin t*cos t/ (cos t - (sin t)^2*cos t)/sin t = 
 = 2sin t*cos t/ (cos t (1- (sin t)^2)/sin t =
= 2sin t/  (1- (sin t)^2)/sin t = 
= 2(sin t)^2/  (1- (sin t)^2) =
=2(sin t)^2/  (1- (sin t)^2)=
= 2(sin t)^2/  ((sin t)^2+(cos t)^2- (sin t)^2)=
 =2(sin t)^2/   (cos t)^2 = 2 (tg t)^2