I want to solve the following:

Prove that

$$\displaystyle \int\_R \sin^{n-2}\phi\_1 \sin^{n-3}\phi\_2\cdots\sin \phi\_{n-2} d\theta d\phi\_1\cdots d\phi\_{n-2} = \frac{2\pi^{n/2}}{\Gamma(n/2)}$$

where $ R=[0,2\pi] \times [0,\pi]^{n-2}$. Hint: Compute $ \int\_{\mathbb R^n}e^{-|x|^2}dx$ in spherical coordinates.

So I am having problems to calculate the latter integral in spherical coordinates because I dont know how to integrate (in finite steps) $sin^{n}(x)$ and I dont know how this results to be a division of integrals.Can you help me to solve this please?, Thanks a lot in advance :)