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Question Number 180621 by Noorzai last updated on 14/Nov/22

Answered by cortano1 last updated on 14/Nov/22

$$\frac{\mathrm{m}}{\mathrm{n}}=\frac{2\cos \left(\frac{\alpha +\beta }{2}\right)\cos \left(\frac{\alpha -\beta }{2}\right)}{2\sin \left(\frac{\alpha +\beta }{2}\right)\cos \left(\frac{\alpha -\beta }{2}\right)}$$$$\Rightarrow \tan \left(\frac{\alpha +\beta }{2}\right)=\frac{\mathrm{n}}{\mathrm{m}}$$$$\Rightarrow \sin (\alpha +\beta )=2\sin \left(\frac{\alpha +\beta }{2}\right)\cos \left(\frac{\alpha +\beta }{2}\right)$$$$=\frac{2\tan \left(\frac{\alpha +\beta }{2}\right)}{1+\tan {}^2\left(\frac{\alpha +\beta }{2}\right)}=\frac{\frac{2\mathrm{n}}{\mathrm{m}}}{1+\frac{{\mathrm{n}}^2}{{\mathrm{m}}^2}}=\frac{2\mathrm{mn}}{{\mathrm{m}}^2+{\mathrm{n}}^2}$$

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