I'd like it explained through the unit circle as I find trig identities easier much easier to understand in this manner.
EDIT: I know you have to apply the identity $\sin(x)=\cos(90-x)$, but I'm wondering how i'd visualise all this on the unit circle?
Best Answer
Using complex numbers and exponential form perhaps help (at least algebraically) to digest these trigonometric addition formulas:
All we have to know is $\cos a+i\cdot\sin a=e^{ai}$ for any $a\in\Bbb R$, and that $i^2=-1$, and that $e^{x+y}=e^x\cdot e^y$ for any $x,y\in\Bbb C$. Then calculate both sides of $e^{(a+b)i}=e^{ai}e^{bi}$.
If you prefer, instead, you can use the matrices of rotation: $$R_a:=\pmatrix{\cos a&-\sin a\\ \sin a &\cos a}$$ and use matrix multiplication to verify the identities, knowing that $$R_{a+b}=R_a\cdot R_b \ .$$