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Question Number 91038 by  M±th+et+s last updated on 27/Apr/20

$$ifsin\left(\frac{\alpha }{2}\right)=\frac{4}{5}$$$$andcos\left(\frac{\beta }{2}\right)=\frac{3}{5}$$$$prove$$$$sin\left(\alpha \right)=cos\left(\beta \right)$$

Commented by jagoll last updated on 27/Apr/20

$$\sin \alpha =2\sin \left(\frac{\alpha }{2}\right)\cos \left(\frac{\alpha }{2}\right)$$$$=2\times \frac{4}{5}\times \frac{3}{5}=\frac{24}{25}$$$$\cos \beta =2\sin \left(\frac{\beta }{2}\right)\cos \left(\frac{\beta }{2}\right)$$$$=2\times \frac{3}{5}\times \frac{4}{5}=\frac{24}{25}$$

Answered by $@ty@m123 last updated on 27/Apr/20

$${\sin }^2\frac{\alpha }{2}+\cos {}^2\frac{\beta }{2}=\frac{4^2+3^2}{5^2}$$$${\sin }^2\frac{\alpha }{2}+\cos {}^2\frac{\beta }{2}=1$$$$\Rightarrow \frac{\alpha }{2}=\frac{\beta }{2}$$$$\Rightarrow \alpha =\beta$$

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