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[S.B.]

5)
[tex]\cos ( \frac{5 \pi }{2} - 2 \alpha) = \cos( \frac{4 \pi }{2} + \frac{ \pi }{2} - 2 \alpha )= \cos(2 \pi + ( \frac{ \pi }{2} - 2 \alpha)) = \cos( \frac{ \pi }{2} - 2 \alpha) = \sin 2 \alpha = 2\sin \alpha \cos \alpha[/tex]
[tex]\sin \alpha = - \frac{2 \sqrt{2} }{3} \Rightarrow \cos \alpha= \sqrt{1 - \frac{8}{9} } = \pm \sqrt{ \frac{1}{9} } = \pm \frac{1}{3}[/tex]
[tex]\alpha \in (180 ^\circ;270 ^\circ ) \Rightarrow \cos \alpha < 0 \Rightarrow \cos \alpha = - \frac{1}{3}[/tex]
[tex]\sin 2 \alpha = 2\sin \alpha\cos \alpha = 2.(- \frac{2 \sqrt{2} }{3})(- \frac{1}{3}) \Rightarrow \sin2 \alpha = \frac{2 \sqrt{2} }{9}[/tex]
$$\Rightarrow \cos( \frac{5 \pi }{2} - 2 \alpha) = \frac{2 \sqrt{2} }{9} $$

6)
[tex]\sin \alpha + 2\sin 3 \alpha + \sin 5 \alpha =[/tex]

[tex]= (\sin \alpha + \sin 5 \alpha ) + 2\sin 3 \alpha = 2\sin \frac{ \alpha + 5 \alpha }{2}\cos \frac{ \alpha - 5 \alpha }{2} + 2\sin 3 \alpha =[/tex]

[tex]= 2\sin 3 \alpha \cos(-2 \alpha) + 2\sin 3 \alpha = 2\sin 3 \alpha(\cos 2 \alpha + 1)=[/tex]

[tex]= 2\sin 3 \alpha.2 \cos^{2 } \alpha =[/tex]

[tex]= 4\sin 3 \alpha \cos^{2 } \alpha[/tex]