\(\large\cot 2\alpha+\csc2\alpha=\frac{1}{\tan 2\alpha}+\frac{1}{\sin 2\alpha}=\frac{1-\tan^2\alpha}{2\tan{\alpha}}+\frac{1}{2\sin{\alpha}\cos{\alpha}}\) \(\Large=\frac{(1-\tan^2\alpha)\cos\alpha}{2\sin\alpha}+\frac{1}{2\sin\alpha.\cos\alpha}=\frac{(1-\tan^2\alpha)\cos^2\alpha+1}{2\sin\alpha\cos\alpha\alpha}\) \(\Large=\frac{\cos^2\alpha-\sin^2\alpha+1}{2\sin\alpha\cos\alpha}=\frac{2\cos^2\alpha}{2\sin\alpha\cos\alpha}=\frac{\cos\alpha}{\sin\alpha}=\cot\alpha\)