Question

Доказать тождество: sin (a-b) sin (a+b) =sin^2 a- sin^2 b

Answer 1

$Sin(\alpha-\beta)Sin(\alpha +\beta)=(Sin\alpha Cos\beta-Cos\alpha Sin\beta)(Sin\alpha Cos\beta +Cos\alpha Sin\beta)=Sin^{2}\alpha Cos^{2}\beta+Sin\alpha Cos\beta Cos\alpha Sin\beta -Sin\alpha Cos\beta Cos\alpha Sin\beta-Cos^{2}\alpha Sin^{2}\beta=Sin^{2}\alpha Cos^{2}\beta -Cos^{2}\alpha Sin^{2}\beta=Sin^{2}\alpha(1-Sin^{2}\beta)-(1-Sin^{2}\alpha)Sin^{2}\beta=$$Sin^{2}\alpha-Sin^{2} \alpha Sin^{2}\beta -Sin^{2}\beta+Sin^{2}\alpha Sin^{2}\beta =Sin^{2}\alpha-Sin^{2}\beta\\\\Sin^{2}\alpha-Sin^{2}\beta=Sin^{2}\alpha-Sin^{2}\beta$

Тождество доказано

Answer 2

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