Are formulas known for non-integer powers of trigonometric functions, analgolous to the power reduction identities? Like if you had a root of sine, could you express it in as a finite summation of sine and cosine functions with different inputs?
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That depends on what you understand by different inputs. For instance, defining $\phi(x)=\arcsin\bigl(\sqrt{\sin x}\bigr)$, $0\le x\le\pi$, we have $$ \sqrt{\sin x}=\sin(\phi(x)),\quad0\le x\le\pi. $$ If you mean scalar multiples of $x$, then the answer is no: $\sin(\phi(x))$ is not differentiable at $x=0$ and at $x=\pi$, but a finite sum of sines and cosines of scalar multiples of $x$ is infinitely differentiable.
However, $\sqrt{\sin x}$ can be written as a Fourier series, like: $$ \sqrt{\sin x}=\sum_{n=1}^\infty a_n\sin(n\,x),\quad0\le x\le\pi, $$ for certain coefficients $a_n$.
Julián Aguirre
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No.
The derivative of $\sqrt{\sin x}$ goes to infinity at $0$, which is not possible for a finite summation of sinusoids.
The situation is similar with
$$(1+x)^n$$ that has a finite expansion, while $$\sqrt{1+x}$$ not.