First, expand the Left Hand Side#("LHS")#
#"LHS"=(tan u + cot u)(cos u + sin u)#
#=tanucosu+tanusinu+cotucosu+cotusinu#
Recall that : #tanu=sinu/cosu#
and #" "cotu=cosu/sinu#
#=>"LHS"=sinu/cancel(cosu)*cancel(cosu)+sinu/cosu*sinu+cosu/sinu*cosu+cosu/cancel(sinu)*cancel(sinu)#
#=sinu+sin^2u/cosu+cos^2u/sinu+cosu#
#=sinu+cos^2u/sinu+cosu+sin^2u/cosu#
#=sin^2u/sinu+cos^2u/sinu+cos^2u/cosu+sin^2u/cosu#
#=(sin^2u+cos^2u)/sinu+(cos^2u+sin^2u)/cosu#
Recall again that : #sin^2u + cos^2u=1#
#=>"LHS"=1/sinu+1/cosu#
Recall that : #1/sinu=cscu#
and also #" "1/cosu=secu#
#=>"LHS"=color(blue)(cscu+secu)#
Quad Era Demonstrandum