Finding co-terminals in trig.

Could anybody explain to me how co-terminal angles work. I need it specifically for simplifying trigonometry expressions.

Let's say we have have something like this:
tan(127) +tan(-143) +tan(-143) -tan(-53) <-- Next finding coterminals..

tan(180-127) +[-tan(180-143)] +[-tan(180-143)]-[-tan(180-53)] <-- Here I calculated the coterminals. I did that for all the parts, not sure I have to find for all of them.

If that's is not the case, when do I need to find coterminals and when not? Continuing:

tan(53) -tan(53) -tan(53)] +tan(127)] <-- Originally the last part of the expression was tan(-53). I
guess I should have not found the co-terminal for that.

If true, how can I know when to apply coterminals and when not?

I can understand the basic principle of what a co-terminal is(I googled and found self-explanatory plots that explain it graphically), but I need help understanding how they work when a plot is not available.

I hope anyone can help me! Please provide as much information as possible!

Thank you!

A = tan(127°) +tan(-143°) +tan(-143°) -tan(-53°)
A = tan(180° - 53°) + 2tan(-143°) - (-tan(53°) )
A = tan(53°) - 2tan(143°) + tan(53°)
A = 2tan(53°) - 2tan(90° + 53°)
A = 2tan(53°) - 2(-cot(53°) )
A = 2tan(53°) + 2cot(53°)
A = 2[ tan(53°) + cot(53°) ]
A = 2[ [(sin²53°) + cos²(53°)] / (sin53° cos53°) ]
A = 2[ [1]/(1/2 sin(2*53°) ]
A = -4/sin(106°)
A = -4/sin(90° + 106°)
A = -4/cos(16°)
A = -4sec(16°)