I've got a question from my Physics textbook that, while I intuitively get the answer, I can't figure out how to "prove" it mathematically. (Listed below)

Question

mendicant_bias · Answer

"You drag a trunk of mass m across a level floor using a massless rope that makes an angle theta with the horizontal. Given a kinetic friction coefficient uk, what rope tension is required to move the trunk at constant speed?"

anonymous · Answer

ll help you.. tell me what you got so far?

mendicant_bias · Answer

The answer is gotten by using a horizontal and vertical coordinate system where the sum of all forces in the x-direction are Tcos(theta) - ukn = 0 and in the y direction Tsin(theta) - mg + n = 0 in the y. It might just be the book being "lazy" but they didn't really bother to specify (i'm guessing we're supposed to assume because of the cos/sin attached? That t for each of those equations only applies for the magnitude of t in that direction.

mendicant_bias · Answer

The answer is$$T = \frac{ u _{k}mg }{ \cos \theta - \sin \theta }$$ I understand intuitively that in order to get the tension when it's at an angle that's not 90 degrees to your coordinate system, you're totally going to have to break it into components along your axes. The way you get this answer is solving for n in the y equation and plugging into the x equation. But how, if I wanted to, could I just add the components together directly instead of substituting? I mean, how could I do something like t(x) + t(y) = T.

anonymous · Answer

wait.. what? :O

since the body is not accelerating, the net force acting on it must be zero..!!.. 
so you resolve the tension in two dimensions.. first of all draw a picture..!

anonymous · Answer

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