$\begin{align}  & \operatorname{cosec}\left( \frac{\pi }{4}-u \right)=\frac{1}{\sin \left( \frac{\pi }{4}-u \right)} \\ & \operatorname{cosec}\left( \frac{\pi }{4}-u \right)=\frac{1}{\sin \tfrac{\pi }{4}\cos u-\sin u\cos \tfrac{\pi }{4}} \\ & \operatorname{cosec}\left( \frac{\pi }{4}-u \right)=\frac{1}{\tfrac{\sqrt{2}}{2}\cos u-\tfrac{\sqrt{2}}{2}\sin u} \\ & \operatorname{cosec}\left( \frac{\pi }{4}-u \right)=\frac{1}{\tfrac{\sqrt{2}}{2}(\cos u-\sin u)} \\ & \operatorname{cosec}\left( \frac{\pi }{4}-u \right)=\frac{2}{\sqrt{2}(\cos u-\sin u)} \\ & \operatorname{cosec}\left( \frac{\pi }{4}-u \right)=\frac{\sqrt{2}}{\cos u-\sin u} \\ & \operatorname{cosec}\left( \frac{\pi }{4}-u \right)=\frac{\sqrt{2}}{\cos u-\sin u}\times \frac{\cos u+\sin u}{\cos u+\sin u} \\ & \operatorname{cosec}\left( \frac{\pi }{4}-u \right)=\frac{\sqrt{2}(\cos u+\sin u)}{{{\cos }^{2}}u-{{\sin }^{2}}x} \\ & \operatorname{cosec}\left( \frac{\pi }{4}-u \right)=\frac{\sqrt{2}(\cos u+\sin x)}{\cos 2x} \\ & \operatorname{cosec}\left( \frac{\pi }{4}-u \right)=\sqrt{2}\sec 2x(\cos u+\sin x) \\\end{align}$