tex]...

July 2022 2 6 Report

Помогите!!! Доказать тождество [tex] \frac{Cos( \alpha - \beta )}{Cos( \alpha + \beta) } = \frac{ctg \alpha *ctg \beta +1}{ctg \alpha *ctg \beta -1} [/tex]

nelle987

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$\dfrac{\mathrm{ctg}\,\alpha\,\mathrm{ctg}\,\beta+1}{\mathrm{ctg}\,\alpha\,\mathrm{ctg}\,\beta-1}=\dfrac{\frac{\cos\alpha}{\sin\alpha}\frac{\cos\beta}{\sin\beta}+1}{\frac{\cos\alpha}{\sin\alpha}\frac{\cos\beta}{\sin\beta}-1}=\dfrac{\cos\alpha\cos\beta+\sin\alpha\sin\beta}{\cos\alpha\cos\beta-\sin\alpha\sin\beta}=\dfrac{\cos(\alpha-\beta)}{\cos(\alpha+\beta)}$

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Матов

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$\frac{ctga*ctgb+1}{ctga*ctgb-1}=\frac{\frac{cosa}{sina}*\frac{cosb}{sinb}+1}{\frac{cosa}{sina}*\frac{cosb}{sinb}-1}=\frac{cosacosb+sinasinb}{cosacosb-sinasinb}\\\\ \frac{cos(a-b)}{cos(a+b)} = \frac{cosacosb+sinasinb}{cosacosb-sinasinb}$
Верно

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