[Math] Trigonometric identities with multiplication

Question

Why aren't there Trigonometric identities with multiplication inside the function? For example for $\sin(xy)=?$.

Answer 1

Probably this won't be a complete answer but I'll consider only the case where $y\in \Bbb N$ (we can even use $y\in \Bbb Z$ but we can easily note that if $y$ is negative we have $\sin(-yx)=-\sin(yx)$ and $\cos(-xy)=\cos(xy)$ so we can treat only positive values of $y$):

We have those so called "multiplication formulas":

$$\sin(nx)=\sum_{k=0}^n\binom n k\cos^k(x)\sin^{n-k}(x)\sin[\frac 12(n-k)\pi] $$

$$\cos(nx)=\sum_{k=0}^n\binom n k\cos^k(x)\sin^{n-k}(x)\cos[\frac 12(n-k)\pi] $$

(The others for tangent and so on follow from this)

These formulas are pretty easy to memorize if you now about Newton's binomial expansion, which is very similar, and as @frog said can be proven by the identities:

$\sin x = \frac{\mathrm e^{\mathrm ix}-\mathrm{e}^{-\mathrm ix}}{2\mathrm i}$

$\cos x = \frac{\mathrm e^{\mathrm ix}+\mathrm{e}^{-\mathrm ix}}{2\mathrm i}$

Btw these formulas are not used so much because they can be deduced by iterating the addition formulas (which are easier to learn and have more uses):

$\sin(x\pm y)=\sin(x)\cos(y)\pm \cos(x)\sin(y)$

$\cos(x\pm y)=\cos(x)\cos(y)\mp \sin(x)\sin(y)$