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If $\cos A + \cos B = m$ and $\sin A + \sin B = n$ where $m, n \neq 0$ , then $\sin (A + B)$ is equal to

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a

mnm2+n2

b

2mnm2+n2

c

m2+n22mn

d

mnm+n

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detailed solution

Correct option is B

We have, mn=(cos⁡A+cos⁡B)(sin⁡A+sin⁡B)⇒mn=cos⁡Asin⁡A+sin⁡(A+B)+sin⁡Bcos⁡B⇒2mn=sin⁡2A+sin⁡2B+sin⁡(A+B)⇒2mn=2sin⁡(A+B)cos⁡(A−B)+sin⁡(A+B) ... (i)Also, we have m2+n2=2+2cos⁡(A−B) ... (ii)From (i) and (ii), we havesin⁡(A+B)=2mnm2+n2

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