find tan theta if 90 degrees < theta < 180 degrees and csc theta = 5/4

cosec \theta = \frac54

That means that sin \theta = \frac45

Remember sin \theta = \frac{1}{cosec \theta

Then, draw a triangle, the angle theta and the sides. You can find the third side through Pythagoras' theorem, or if you know your pythagorean triples, 3, 4, 5.

Then you'll immediately see what is tan theta. However, be careful, for the theta is larger than 90 degrees. You'll have to look for the quadrant that corresponds to the problem.