sinx+cosx=1/sinx скажите решение пожалуйста!!!

Решите задачу:

$\sin x + \cos x = \frac{1}{\sin x} \ | \ * \sin x, \ \sin x \ne 0\\\\ \sin^2 x + \sin x \cos x = 1\\\\ \sin^2 x + \sin x \cos x = \sin^2 x + \cos^2 x\\\\ \sin x \cos x - \cos^2 x = 0\\\\ \cos x (\sin x - \cos x) = 0\\\\ 1) \ \cos x = 0, \ x = \frac{\pi}{2} + \pi n, \ n \in \mathtt{Z}\\\\ 2) \ \sin x - \cos x = 0, \ \sin x = \cos x, \ x = \frac{\pi}{4} + \pi n, \ n \in \mathtt{Z}$