Odpowiedź :
[tex]\cos2\alpha=\cos^2\alpha-\sin^2\alpha=1-2\sin^2\alpha=2\cos^2\alpha-1[/tex]
Uwaga: Ponieważ kąt 22,5° jest ostry, więc sinus i cosinus będą dodatnie.
a)
[tex]\cos45^\circ=2\cos^222,5^\circ-1\\\frac{\sqrt2}{2}=2\cos^222,5^\circ-1\ |*2\\\sqrt2=4\cos^222,5^\circ-2\\\sqrt2+2=4\cos^222,5^\circ\ |:4\\\cos^222,5^\circ=\frac{\sqrt2+2}{4}\ |\sqrt{}\\\cos22,5^\circ=\frac{\sqrt{\sqrt2+2}}{2}[/tex]
b)
[tex]\cos45^\circ=1-2\sin^222,5^\circ\\\frac{\sqrt2}{2}=1-2\sin^222,5^\circ\ |*2\\\sqrt2=2-4\sin^222,5^\circ\\4\sin^222,5^\circ=2-\sqrt2\ |:4\\\sin^222,5^\circ=\frac{2-\sqrt2}{4}\ |\sqrt{}\\\sin22,5^\circ=\frac{\sqrt{2-\sqrt2}}{2}[/tex]
