[tex]mam ~~dany~~kat~~\alpha ~~gdzie ~~\alpha \in(0^\circ ; 90^\circ)~~\Rightarrow ~~\alpha ~~ jest~katem~~ostrym~\\\\cos\alpha = \sqrt{2} -1\\\\oblicze~~sin\alpha ~~korzystam ~~z~~jedynki~~trygonometrycznej:\\\\sin^{2} \alpha + cos^{2} \alpha =1~~ \land ~~cos\alpha = \sqrt{2} -1\\\\sin^{2} \alpha + (\sqrt{2} -1)^{2} =1\\\\sin^{2} \alpha +2 -2\sqrt{2} +1=1\\\\sin^{2} \alpha =2\sqrt{2} -2~~\land ~~~\alpha \in(0^\circ ; 90^\circ)\\\\sin\alpha = \sqrt{2\sqrt{2} -2} \\\\[/tex]
[tex]sin\alpha = \sqrt{2(\sqrt{2} -1) } \\[/tex]
[tex]zad.a\\2sin^{2} - 2 = ? ~~ \land ~~ sin^{2} \alpha =2\sqrt{2} -2\\\\~~~~~~~~~~~~~~~~~~~~~~~~~\Downarrow \\\\2\cdot (2\sqrt{2} -2) - 2 = 4\sqrt{2} -4-2=4\sqrt{4} -6\\\\[/tex]
[tex]zad.b\\\\ctg\alpha =\frac{cos\alpha }{sin\alpha } \\\\\sqrt{2} ctg^{2} \alpha \cdot sin^{2} \alpha =\sqrt{2}\frac{cos^{2} \alpha }{sin^{2} \alpha } \cdot sin^{2} \alpha = \sqrt{2} cos^{2} \alpha =?\\\\ \sqrt{2} cos^{2} \alpha~~\land~~ cos^{2} \alpha =\sqrt{2} - 1\\\\~~~~~~~~~~~~~~~~~~~~~~~~\Downarrow \\\\\sqrt{2} (\sqrt{2} -1)= 2 - \sqrt{2}[/tex]
