\(a) \, \, {{1 - \cos x + \cos 2x} \over {\sin 2x - {\mathop{\rm s}\nolimits} {\rm{in x}}}} = {{1 + \cos 2x - \cos x} \over {2\sin x\cos x - {\mathop{\rm sinx}\nolimits} }} \)\(= {{\cos x(2\cos x - 1)} \over {{\mathop{\rm s}\nolimits} {\rm{inx}}(2\cos x - 1)}} = \cot x\)
\( b) \, \, {{{\mathop{\rm sinx}\nolimits} + sin{x \over 2}} \over {1 + \cos x + \cos {x \over 2}}}\)
\(= {{2\sin {x \over 2}\cos {x \over 2} + \sin {x \over 2}} \over {2{{\cos }^2}{x \over 2} + \cos {x \over 2}}}\)
\(= {{\sin {x \over 2}(2\cos {x \over 2} + 1)} \over {\cos {x \over 2}(2\cos {x \over 2} + 1)}}\)
\(=\tan {x \over 2}. \)
\(c) \, \, {{2\cos 2x - \sin 4x} \over {2\cos 2x + \sin 4x}}\)
\(= {{2\cos 2x - 2\sin2 x\cos 2x} \over {2\cos 2x + 2\sin 2x\cos 2x}}\)
\(= {{1 - \sin 2x} \over {1 + \sin 2x}}\)\(= {{1 - \cos ({\pi \over 2} - 2x)} \over {1 + \cos ({\pi \over 2} - 2x)}}\)
\(= {{2{{\sin }^2}({\pi \over 4} - x)} \over {2{{\cos }^2}({\pi \over 4} - x)}}\)
\(= {\tan ^2}({\pi \over 4} - x) \)
\(d) \, \, \tan x - \tan y\)
\(= {{{\mathop{\rm sinx}\nolimits} } \over {{\mathop{\rm cosx}\nolimits} }} - {{\sin y} \over {\cos y}}\)
\(= {{\sin {\rm{x}}\cos y - \cos x\sin y} \over {\cos x\cos y}}\)
\(= {{\sin (x - y)} \over {\cos x\cos y}}.\)
