1.
csc/(cot+tan)
=1/sin/(sin/cos+cos/sin)
=1/sin/(1/sin cos)
=cos
2.
tan+cos/(1+sin)
=sin/cos+cos/(1+sin)
=(sin+sin²+cos²)/(cos(1+sin))
=(1+sin)/(cos(1+sin))
=sec
3.
tan+sec=sin/cos+1/cos=(1+sin)/cos
sec-cos+tan=1/cos-cos+sin/cos=(1-cos²+sin)/cos=sin(1+sin)/cos
(tan+sec)/(sec-cos+tan)=1/sin=csc