Add or subtract as indicated, and then simplify as much as possible. Leave your answer in terms of sines and cosines only, \( \frac{\cos (x)}{\sin x}+\frac{\sin (x)}{\cos (x)}=\square \)

To add the two fractions, first find a common denominator, which is \(\sin x \cos x\): \[ \frac{\cos (x)}{\sin x} + \frac{\sin (x)}{\cos (x)} = \frac{\cos^2 (x)}{\sin x \cos x} + \frac{\sin^2 (x)}{\sin x \cos x} \] Now combine the numerators: \[ = \frac{\cos^2 (x) + \sin^2 (x)}{\sin x \cos x} \] Using the Pythagorean identity, \(\cos^2 (x) + \sin^2 (x) = 1\): \[ = \frac{1}{\sin x \cos x} \] This can be rewritten using the double angle identity: \[ = \frac{2}{\sin(2x)} \] So, the simplified answer is \(\frac{2}{\sin(2x)}\).