Solve f^n=sin(x)cos(x)

$$f ^ { n } = \sin ( x ) \cos ( x )$$

Solve for f (complex solution)

$$f=\left(\frac{|\sin(2x)|}{2}\right)^{\frac{Re(n)-iIm(n)}{\left(Re(n)\right)^{2}+\left(Im(n)\right)^{2}}}e^{\frac{Im(n)arg(\sin(2x))-2\pi n_{1}iRe(n)-2\pi n_{1}Im(n)+iRe(n)arg(\sin(2x))}{\left(Re(n)\right)^{2}+\left(Im(n)\right)^{2}}}$$

$$n_{1}\in \mathrm{Z}$$

Solve for n (complex solution)

$$\left\{\begin{matrix}n=\frac{\ln(\sin(2x))-\ln(2)+2\pi n_{1}i}{\ln(f)}\text{, }n_{1}\in \mathrm{Z}\text{, }&\nexists n_{2}\in \mathrm{Z}\text{ : }x=\frac{\pi n_{2}}{2}\text{ and }f\neq 1\text{ and }f\neq 0\\n\in \mathrm{C}\text{, }&\exists n_{4}\in \mathrm{Z}\text{ : }x=\pi n_{4}-\frac{i\ln(\sqrt{3}i+2i)}{2}\text{ or }\exists n_{3}\in \mathrm{Z}\text{ : }x=\pi n_{3}-\frac{i\ln(-\sqrt{3}i+2i)}{2}\end{matrix}\right.$$

Solve for n

$$\left\{\begin{matrix}n=\frac{\ln(\sin(2x))-\ln(2)}{\ln(f)}\text{, }&\exists n_{2}\in \mathrm{Z}\text{ : }\left(x>\pi n_{2}\text{ and }x<\pi n_{2}+\frac{\pi }{2}\right)\text{ and }f\neq 1\text{ and }f>0\\n>0\text{, }&f=0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }x=\frac{\pi n_{1}}{2}\end{matrix}\right.$$