sinx/cosx = tanx

I’m not sure what you meant by cosx.cscx. Did you mean to multiply or divide, or what?

I’m guessing multiply for now.

cosx * (1/sinx) = cosx/sinx = tanx.

SecA/CscA: Sec is 1/cos, and csc is 1/sin. So sec/csc=(1/cos)/(1/sin) = sin/cos = tan

This next one would be good to draw in a set of coordinate axes. If csc is positive, then both the opposite and hypotenuse must be either positive or negative. (They are the same.) Tangent is basically slope, and if slope is negative it must be in either II or IV. Then the opposite must be negative, so it is in the IV quadrant. (I’m not 100% sure about this one)

Um… I’m going to guess that these are being multiplied together.

Cosx * (sinx/cosx) * (1/sinx). It’s clearer if you actually write it vertically instead of having the slash for a divide sign. Well, the cosines cancel and the sines cancel, and you end up with 1.