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Find the value of (sin 5degree - sin 67degree +sin 77degree - sin 139degree + sin 149degree).
Hi Arnab,
Use the identity SinA-SinB = 2Sin(A-B)/2Cos(A+B)/2.
So sin77-sin67 = 2sin5cos72
And sin149-sin139 = 2sin5cos144 = -2sin5cos36 -----(since cos144 = cos(180-36) = -cos36)
So this sum is 2sin5(cos72-cos36) = 4sin5(-sin18sin54) = -4sin5(sin18cos36)
We know sin18 = (√5-1)/4, and cos36 = (√5+1)/4
So sin18cos36 = (√5-1)(√5+1)/16 = 1/14
And hence -4sin5(sin18cos36) = -sin5
So the required sum finally is sin5 - sin5 = 0
Hence 0 is the value of the sum in the Question.
Hope this helps.
Best Regards,
Ashwin (IIT Madras).
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