If sin A + sin B = x and cos A + cos B = y,th | vedclass

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(A) Given: $\sin A + \sin B = x$  $(1)$ and $\cos A + \cos B = y$  $(2)$. Using sum-to-product formulas: $2 \sin(\frac{A+B}{2}) \cos(\frac{A-B}{2}) =

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$\frac{2xy}{x^2 + y^2}$

Vedclass · Answer

(A) Given: $\sin A + \sin B = x$  $(1)$ and $\cos A + \cos B = y$  $(2)$. Using sum-to-product formulas: $2 \sin(\frac{A+B}{2}) \cos(\frac{A-B}{2}) = x$  $(3)$ $2 \cos(\frac{A+B}{2}) \cos(\frac{A-B}{2}) = y$  $(4)$ Dividing $(3)$ by $(4)$: $	an(\frac{A+B}{2}) = \frac{x}{y}$. We know that $\sin(A+B) = \frac{2 	an(\frac{A+B}{2})}{1 + 	an^2(\frac{A+B}{2})}$. Substituting the value: $\sin(A+B) = \frac{2(x/y)}{1 + (x/y)^2} = \frac{2x/y}{(y^2 + x^2)/y^2} = \frac{2xy}{x^2 + y^2}$.

If $\sin A + \sin B = x$ and $\cos A + \cos B = y$,then $\sin(A + B) = $

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