SOLUTION: If 0<A<90 degrees, prove geometrically that sec(A-180°) = -secA

Draw the two right triangles congruent. 
The angle indicated by the counter-clockwise red arc is A.
The angle indicated by the clockwise green arc is A-180°.
The values of x and y are positive, and r is ALWAYS taken positive.



 

So  proved geometrically.

Edwin