Math, asked by NitishKdhiran, 2 days ago

If sin α + sin β = a and cos α + cos β = b then show that sin (α+β) =
2ab
/a²+b²​

Answers

Answered by bramarambikavns
1

Answer:

\frac{2ab}{a^{2} + b^{2} } = sin(\alpha + \beta )

Step-by-step explanation:

sin (\alpha + \beta ) = \frac{2ab}{a^{2} + b^{2} } \\a = sin \alpha  + sin \beta = 2sin (\frac{\alpha + \beta }{2} ) . cos (\frac{\alpha - \beta }{2} )\\b = cos \alpha  + cos \beta = 2cos (\frac{\alpha + \beta }{2} ) . cos (\frac{\alpha - \beta }{2} )\\ab = 4sin (\frac{\alpha + \beta }{2} ) . cos (\frac{\alpha + \beta }{2} ) . cos (\frac{\alpha -\beta }{2} )\\= 2sin(\alpha +\beta ). cos^{2}(\frac{\alpha -\beta }{2} ) \\\\a^{2} + b^{2} = 4cos^{2}(\frac{\alpha - \beta }{2} )   \\

\frac{2ab}{a^{2} + b^{2} } = sin(\alpha + \beta )

proved.

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