Докажите тождество 2sin^2(α)/tg2α*tgα=cos^2(α)-sin^2(α)

Докажем тождество:

2 * sin^2 a/(tg (2 * a) * tg α) = cos^2 α - sin^2 α;  

2 * sin^2 a/(tg (2 * a) * sin α/cos a) = cos^2 α - sin^2 α;  

2 * sin^2 a/((tg (2 * a) * sin α)/cos a) = cos^2 α - sin^2 α;  

2 * sin^2 a * cos a/(tg (2 * a) * sin a) = cos^2 a - sin^2 a; 

2 * sin a * cos a/(tg (2 * a) * 1) = cos^2 a - sin^2 a;  

2 * sin a * cos a/(sin (2 * a)/cos (2 * a)) = cos^2 a - sin^2 a;  

sin (2 * a)/((sin (2 * a)/cos (2 * a)) = cos^2 a - sin^2 a;  

sin (2 * a) * cos (2 * a)/sin (2 * a) = cos^2 a - sin^2 a; 

cos (2 * a) = cos^2 a - sin^2 a; 

cos^2 a - sin^2 a = cos^2 a - sin^2 a; 

Тождество верно.