А) ctgα+ sinα / 1+cosα = (cosα/sinα + sinα) / (1+cosα)
= (cosα + sin^2α) / (sinα(cosα + 1))
= (1 + sin^2α) / (1 + cosα)
= cos^2α / cosα
= cosα

Б) sin(30+α)-cos(60+α) / sin(30+α)+cos(60+α) = (sin30cosα + cos30sinα - cos60cosα + sin60sinα) / (sin30cosα + cos30sinα + cos60cosα + sin60sinα)
= ((1/2)(cosα) + (sqrt(3)/2)(sinα) - (1/2)(cosα) - (sqrt(3)/2)(sinα)) / ((1/2)(cosα) + (sqrt(3)/2)(sinα) + (1/2)(cosα) + (sqrt(3)/2)(sinα))
= 0 / (2cosα + 2sqrt(3)sinα)
= 0

В) cos7α+cosα / sin7α-sinα = 2cos4αcos3α / 2sin4αcos3α
= cos4α / sin4α
= tan4α