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If $0 < x, y < π and \cos x + \cos y - \cos (x + y) = \frac{3}{2}$ then $\sin x + \cos y$ is equal to

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a

12

b

1+32

c

32

d

1-32

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detailed solution

Correct option is B

2cos⁡x+y2cos⁡x−y2−2cos2⁡x+y2−1=32⇒ 2cos⁡x+y2cos⁡x−y2−2cos2⁡x+y2=12⇒4cos⁡x+y2cos⁡x−y2−4cos2⁡x+y2=1=cos2⁡x−y2+sin2⁡x−y2⇒4cos2⁡x+y2+cos2⁡x−y2−4cos⁡x+y2cos⁡x−y2+sin2⁡x−y2=0⇒cos⁡x−y2−2cos⁡x+y22+sin2⁡x−y2=0⇒ sin⁡x−y2=0⇒x=y⇒ cos⁡x−y2=2cos⁡x+y2 Now. put x=y ⇒ cos⁡x=12=cos⁡y∴ sin⁡x+cos⁡y=32+12=3+12

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