Tg2a (1-tg^ 2a)/(1+tg^ 2a)=sin2a доказать тождествос решением

$\mathtt{\frac{tg2a(1-tg^2a)}{1+tg^2a}=sin2a;~\frac{tg2a(1-\frac{sin^2a}{cos^2a})}{tg2a(1+\frac{sin^2a}{cos^2a})}=\frac{sin2a}{tg2a};~\frac{\frac{cos^2a-sin^2a}{cos^2a}}{\frac{cos^2a+sin^2a}{cos^2a}}=\frac{sin2a}{\frac{sin2a}{cos2a}}=cos2a;}\\\\\mathtt{\frac{cos^2a-sin^2a}{cos^2a+sin^2a}=cos2a;~\frac{cos^2a-sin^2a}{1}=cos2a;~cos^2a-sin^2a=cos2a}$

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