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प्रश्न
If \[\sin \alpha + \sin \beta = a \text{ and } \cos \alpha + \cos \beta = b\] , prove that
(i)\[\sin \left( \alpha + \beta \right) = \frac{2ab}{a^2 + b^2}\]
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उत्तर
The given equations are \[\sin \alpha + \sin \beta = a \text{ and } \cos \alpha + \cos \beta = b\]
(i) \[\because \text{ sin } C + \text{ sin } D = 2\sin\frac{C + D}{2}\cos\frac{C - D}{2}\]
\[ \therefore 2\sin\frac{\alpha + \beta}{2}\cos\frac{\alpha - \beta}{2} = a . . . (1)\]
Now, using the identity
\[ \Rightarrow \sin\left( \alpha + \beta \right) = \frac{2 \times \frac{a}{b}}{1 + \frac{a^2}{b^2}} = \frac{2ab}{a^2 + b^2}\]
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