Get recurrent formula for integral $I_n=\frac{1}{\sin^n(x)}$ The first thing that I could notice is that we can do so
$$I_n - I_{n-2} = \int \frac{1 - \sin^2 x}{\sin^n x} dx = \int \frac{\cos^2 x}{\sin^n x} dx$$
(Sometimes I will skip putting $dx$ for contraction)
However when I was trying to continue integrating in this form, it seemed to me as even harder problem and I decided to change approach and try integrating by parts.
$$I_n = \int \frac{1}{\sin^n x} = \int \frac{1}{\sin x \cdot \sin^{n - 1}x}$$
This one was a bit despeate, I know. So, I decided on putting this idea aside either.
So, do you have any ideas on how to approach this problem? Thank you in advance.

$$
I_n = \int \dfrac{\sin x}{\sin^{n+1} x}dx = - \dfrac{\cos x}{\sin^{n+1} x} - (n+1)\int \dfrac{\cos^2 x}{\sin^{n+2} x}dx = - \dfrac{\cos x}{\sin^{n+1} x} - (n+1)(I_{n+2}-I_n)
$$
