TITLE: What is the purpose of the compound angle identity in trigonometry?
QUESTION [7 upvotes]: This may be a silly question, but one that I am confused about nonetheless.
With regards to the compound trig identities such as $\cos(A+B)=\cos A\cos B - \sin A\sin B$ etc., I'd like to know why they are used. What's the purpose? Surely, one would ask themselves that if we can just add $A$ and $B$ together and call it $C$ (degrees or radians) then we could just find cosine of $C$...?
I know there's more to it, but that's where I'd love to get your help.
Thank you! :)

REPLY [2 votes]: The purpose of these equations are to enable a student, mathematician, engineer, etc... to solve equations.  You will see it alot in the following classes: Integral Calculus, Multivariable and Vector Calculus, Differential Equations, etc....
A really good example that is important in signal processing is the Fourier Series.  It's a form of interpolation which is used to build waveforms.
http://mathworld.wolfram.com/FourierSeries.html
http://en.wikipedia.org/wiki/Fourier_series
As someone else mentioned, you could use a calculator to do this, but for pencil and paper work without a calculator, these identities come in handy to solve such equations.  There are many such identities in all of mathematics, but no one person can know them all, not even the one's who have PHD degrees.  The ones they teach in school are the most popular/useful ones.