2) \( \frac{\sin (u+v)}{\cos (u) \cos (v)}=\tan (u)+\tan (v) \)

Did you know that the equation \( \frac{\sin (u+v)}{\cos (u) \cos (v)}=\tan (u)+\tan (v) \) is a delightful mix of trigonometric identities? It stems from the sine addition formula, where \( \sin(u+v) = \sin(u)\cos(v) + \cos(u)\sin(v) \). When you divide both sides by \( \cos(u)\cos(v) \), you get to see the beauty of how the basic trigonometric functions interact! In practice, this identity helps simplify calculations in fields such as physics and engineering, particularly in wave mechanics or signal processing. By using this relation, engineers can break down complex waveforms into simpler components, making it easier to analyze and design circuits or signals. So, next time you’re tackling trig problems, remember that these identities are not just academic—they're tools you can wield in the real world!