csc x + tan2 x csc x = 1/sin x + (sin x)2/(cos x)2 · 1/sin x (express in terms of sin x and cos x)
= 1/sin x + sin x / cos2 x (cancel sin x)
= (cos2 x + sin2 x)/(sin x cos2 x) (common denominator)
= 1/(sin x cos2 x) (Pythagorean identity)
= (1/sin x)(1/cos2 x) (break up the fraction as a product)
= csc x sec x (use basic identities to rewrite)
I hope this will be of use to you.