a) \({{2\sin 2\alpha - \sin 4\alpha } \over {2\sin 2\alpha + \sin 4\alpha }} \)

\(= {{2\sin 2\alpha - 2\sin 2\alpha .cos2\alpha } \over {2\sin 2\alpha + 2\sin 2\alpha .cos2\alpha }}\)

\(= {{1 - \cos 2\alpha } \over {1 + \cos 2\alpha }} \)

\(= {{2{{\sin }^2}\alpha } \over {2{{\cos }^2}\alpha }} =\tan^2\alpha \)

b) \(\tan \alpha \left({{1 + {{\cos }^2}\alpha } \over {\sin \alpha }} - \sin \alpha\right ) \)

\(= {{\sin \alpha } \over {\cos \alpha }}\left({{1 + {{\cos }^2}\alpha - {{\sin }^2}\alpha } \over {\sin \alpha }}\right) \)

\(= {{\sin \alpha } \over {\cos \alpha }}.{{2{{\cos }^2}\alpha } \over {\sin \alpha }} \)

\(= 2\cos \alpha \)

c) \({{\sin ({\pi  \over 4} - \alpha ) + \cos ({\pi  \over 4} - \alpha )} \over {\sin ({\pi  \over 4} - \alpha ) - \cos ({\pi  \over 4} - \alpha )}}\)

\(= {{\tan \left({\pi \over 4} - \alpha \right) + 1} \over {\tan\left({\pi \over 4} - \alpha \right) - 1}} \)

\(= \left({{\tan {\pi \over 4} - \tan \alpha } \over {1 + \tan {\pi \over 4}.\tan \alpha }} + 1\right):\left({{\tan {\pi \over 4} - \tan \alpha } \over {1 + \tan {\pi \over 4}.\tan \alpha }} - 1\right) \)

\(= \left({{1 - \tan \alpha + 1 + \tan \alpha } \over {1 + \tan \alpha }} \right):\left({{1 - \tan \alpha - 1 - \tan \alpha } \over {1 + \tan \alpha }} \right) \)

\(= {{ - 1} \over {\tan \alpha }} = - \cot \alpha \) 

d) \({{\sin 5\alpha  - \sin 3\alpha } \over {2\cos 4\alpha }} \)

\(= {{2\cos {{5\alpha  + 3\alpha } \over 2}\sin {{5\alpha  - 3\alpha } \over 2}} \over {2\cos 4\alpha }} \)

\(= \sin \alpha \)