Find all solutions of the equation in the interval [0,2pi ) csc2x-sqrt (2)=0 Write your answer in radians in terms of pi If there is more than one solution , separate them with commas.

To find all solutions of the equation $\csc 2x - \sqrt{2} = 0$ in the interval $[0, 2\pi)$, we first solve the equation for $\csc 2x$:<br /><br />$\csc 2x = \sqrt{2}$<br /><br />Since $\csc 2x = \frac{1}{\sin 2x}$, we can rewrite the equation as:<br /><br />$\frac{1}{\sin 2x} = \sqrt{2}$<br /><br />Taking the reciprocal of both sides, we get:<br /><br />$\sin 2x = \frac{1}{\sqrt{2}}$<br /><br />Simplifying further, we have:<br /><br />$\sin 2x = \frac{\sqrt{2}}{2}$<br /><br />The solutions for $\sin 2x = \frac{\sqrt{2}}{2}$ in the interval $[0, 2\pi)$ are:<br /><br />$2x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$<br /><br />Dividing each of these by 2, we get:<br /><br />$x = \frac{\pi}{8}, \frac{3\pi}{8}, \frac{5\pi}{8}, \frac{7\pi}{8}$<br /><br />Therefore, the solutions to the equation $\csc 2x - \sqrt{2} = 0$ in the interval $[0, 2\pi)$ are:<br /><br />$x = \frac{\pi}{8}, \frac{3\pi}{8}, \frac{5\pi}{8}, \frac{7\pi}{8}$