Докажите тождества а) [latex] \frac{cos2a}{1+sin2a} = \frac{1-tga}{1+ tga} [/latex] б)[latex]2cos(45 ^{\circ} +a)*cos(45 ^{\circ} - a) = cos 2a[/latex]

[latex]a)\ \frac{\cos2 \alpha }{1+\sin2 \alpha }=\frac{\cos^2 \alpha -\sin^2 \alpha }{\sin^2 \alpha +\cos^2 \alpha +2\sin \alpha \cos \alpha }=\frac{(\cos \alpha -\sin \alpha )(\cos \alpha +\sin \alpha )}{(\cos \alpha +\sin \alpha )^2}=\\\\=\frac{\cos \alpha -\sin \alpha }{\cos \alpha +\sin \alpha }=\frac{\cos \alpha (1-\frac{\sin \alpha }{\cos \alpha })}{\cos \alpha (1+\frac{\sin \alpha }{\cos \alpha })}=\frac{1-tg \alpha }{1+tg \alpha };[/latex] [latex]b)\ 2\cos(45^\circ+ \alpha )\bullet\cos(45^\circ- \alpha )=\\\\2\bullet\frac{1}{2}(\cos((45^\circ+ \alpha) -(45^\circ- \alpha ))+\cos(45^\circ+ \alpha +45^\circ- \alpha ))=\\\\\cos2 \alpha +\cos90^\circ=\cos2 \alpha +0=\cos2 \alpha .[/latex]