Question

доведіть тотожність ​

Answer 1

Доказать тождество $\underline{ \dfrac{\tan {\alpha}}{1+\tan{\alpha}} + \dfrac{\tan {\alpha}}{1-\tan{\alpha}} = \tan{2\alpha}}$

Ответ:

$\dfrac{\tan {\alpha}}{1+\tan{\alpha}} + \dfrac{\tan {\alpha}}{1-\tan{\alpha}} = \tan{2\alpha}$

Приводим к общему знаменателю первую часть уравнения:

$\dfrac{(1-\tan{\alpha})*\tan{\alpha}+(1+\tan{\alpha})*\tan{\alpha}}{(1+\tan{\alpha})*(1-\tan{\alpha})} = \tan{2\alpha}$

Упрощаем:

$\dfrac{\tan{\alpha}-\tan{\alpha}^2+\tan {\alpha}+\tan{\alpha}^2} {1-\tan{\alpha}^2}= \tan{2\alpha}\\\\\\\dfrac{2\tan{\alpha}} {1-\tan{\alpha}^2}= \tan{2\alpha}\\\\\\\underline {\tan{2\alpha}= \tan{2\alpha}}$

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