Is there a visual proof of the addition formula for $\sin^2(a+b)$ ? The visual proof of the addition formula for $\sin(a+b)$ is here :
Also it is easy to generalize (in any way: algebra , picture etc) to an addition formula for $\sin^n(a+b)$ where $n$ is a given positive integer ?
4 COMMENTS :
$1)$ I prefer the addition formula's to have as little sums as possible. I assume this is equivalent to allowing and preferring large power of $\sin$ and $\cos$ ; e.g. $\sin^4(a+b)=$ expression involving $\sin^2$, $\sin^4$ and $\cos^4$ and no other powers of $\sin$ or $\cos$.
In one of the answers, the poster just used the binomium. That works nice, but Im not sure I like the output of that answer. We get $n$ sums for $sin^n(a+b)$ and I assume we can do better if we allow powers of $\sin$ and $\cos$.
I could be wrong ofcourse.
$2)$ I bet against the existance of visual proofs for $\sin^n(a+b)$ for $n>1$. But I could be wrong.
$3)$ Not trying to insult the answers and comments but I am skeptical about the use of complex numbers (and $\exp$) to solve this issue EFFICIENTLY. I know Euler's formula for $\exp(i x)$ but still. I could be wrong about this too ofcourse.
$4)$ My main intrest is in the $\sin^2$ case. I assume it has many forms. Can the addition formula for $\sin^2$ be expressed by $sin^2$ only ? I think so. (One of the reason I think so is because $\cos^2$ can be rewritten.)