sec(a+10)=csc(2a+10),
sec(a+10)=sec(90-(2a+10)), so:
a+10=90-2a-10+360n, where n is an integer.
3a=70+360n, a=70/3+120n degrees is the general solution.
But csc(2a+10)=csc(180-(2a+10))=csc(170-2a).
sec(a+10)=csc(90-(a+10)), so:
90-(a+10)=170-2a+360n,
80-a=170-2a+360n,
a=170-80+360n=90+360n.
Let's check the results: (n=0) a=70/3:
sec(70/3+10)=csc(140/3+10),
sec(100/3)=csc(170/3)=1.1969 approx.
sec(100)=csc(190)=-5.7604 approx.
Therefore a=70/3+360n or 90+360n.