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**Homework Statement:**This is not a homework question.

I am trying to understand why we spend so much time studying trig identities.

**Homework Equations:**As far as I understand, the two basic trig functions (sin and cos ) show the relationship between the sides of a right angle triangle in a unit circle. We care about them because we can apply this knowledge in many engineering and physics situations.

What I don't get is why we care so much about identities like

Why should I spend time learning and remembering something like ## cos u . sin v = \frac {1}{ 2} [sin(u + v) − sin(u − v)] ##

Here is how I tried to understand ## (sin u + cos v )^2 = sin^2 u +cos^2 v + 2 sin u . cos v ## . All I see is sinusoidal functions with different periods and some of them are always positive.

How do I use this knowledge?