HappMatt said:
Also I haven't really seen this expression in precalc but it seems like i have seen it used often on this board. What exactly is its purpose??
Sounds like a bad precalc. class.
The best thing about the complex trig identities at the elementary level is that it make it easy to remember the coresponding real identities.
For example say you forget (unlikely as it may seem perhaps you receive a concusion)
\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)
but remember
e^{a+b}=e^ae^b
and
e^{i x}=\cos(x)+i \sin(x)
so
e^{(x+y)i}=e^{x i}e^{y i}
hence
\cos(x+y)+i \sin(x+y)=(\cos(x)+i \sin(x))(\cos(y)+i \sin(y))
It also makes it easy to remember the other identities and is you also do hyperbolic trig it makes it easy to relate its identities to those of circular trig.
Anyway about identities if you have not got a knack for them by now you should probably focus on slow reliable methods not quick and clever ones.
Here are a few tips
-work with the same functions on both sides
-work with sin and cos only
-write all the arguments in terms of the gcd of all arguments
-reduce higher polynomials to lower with pythag idens
-multiply denominators by congugates
-avoid radicals when possible
-factor
-it can help to combine or separate fractions
Here is a simple example
\frac{2\tan^2(x)+2\tan(x)\sec(x)}{\tan(x)+\sec(x)-1}=\tan(x)+\sec(x)+1
attack the RHS
\frac{\tan(x)+\sec(x)-1}{\tan(x)+\sec(x)-1}(\tan(x)+\sec(x)+1)=\frac{\tan^2(x)+2\sec(x)\tan(x)+sec^2(x)-1}{\tan(x)+\sec(x)-1}
use pythag
(\tan(x)+\sec(x)+1)=\frac{2\tan^2(x)+2\sec(x)\tan(x)}{\tan(x)+\sec(x)-1}
QED
