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If $\sin A + \sin B = a$ and $\cos A + \cos B = b,$ then $\cos (A + B)$

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a2+b2b2−a2

2aba2+b2

b2−a2a2+b2

a2−b2a2+b2

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detailed solution

Correct option is C

We have,a=sin⁡A+sin⁡B,b=cos⁡A+cos⁡B ⇒ a2+b2=2+2cos⁡(A−B) …(i)and, b2−a2=cos⁡2A+cos⁡2B+2cos⁡(A+B)⇒b2−a2=2cos⁡(A+B){cos⁡(A−B)+1}⇒ b2−a2=2cos⁡(A+B)a2+b22 [Using (i)]⇒ cos⁡(A+B)=b2−a2a2+b2

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If sin⁡A+sin⁡B=a and cos⁡A+cos⁡B=b, then cos⁡(A+B)

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If $\sin A + \sin B = a$ and $\cos A + \cos B = b,$ then $\cos (A + B)$