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).