Q.Show that one value of $\left[\dfrac{1 + \sin\frac{\pi}{8} + i\cos\frac{\pi}{8}}{1 + \sin\frac{\pi}{8} - i\cos\frac{\pi}{8}}\right]^{8/3}$ is $-1$.

The expression inside the bracket equals $e^{i\,3\pi/8}$, so its $8/3$-power is $e^{i\pi}=-1$.

Let $\theta=\tfrac{\pi}{8}$ and put $\varphi=\tfrac{\pi}{2}-\theta=\tfrac{3\pi}{8}$.

Then $\sin\theta=\cos\varphi$ and $\cos\theta=\sin\varphi$, so

Numerator $=1+\cos\varphi+i\sin\varphi=2\cos^2\tfrac{\varphi}{2}+2i\sin\tfrac{\varphi}{2}\cos\tfrac{\varphi}{2}=2\cos\tfrac{\varphi}{2}\,e^{i\varphi/2},$

Denominator $=1+\cos\varphi-i\sin\varphi=2\cos\tfrac{\varphi}{2}\,e^{-i\varphi/2}.$

…