Simplification 30pi k2 sin pi k2 2 sin 3 pi k2 120pi k si

Simplification = 30/pi k[2 sin pi k/2 - 2 sin 3 pi k/2] = 120/pi k sin pi k/2

Solution

30/pi*k(2sin(pi k/2)- 2sin (3pi k/2) )=120/pi k(sin pi k/2)

Taking the 2 out from the parenthesis

60/pi k(sin (pi k/2)- sin(3 pi k/2))

Using sin A- sin B= 2 cos ((A+B)/2)sin ((A-B)/2)

60/pi k (2 cos ((pi k/2)+(3 pi k/2)/2)sin((pi k/2)-(3 pi k/2)/2))

120/pi k (cos ((4pi k/2)/2)sin((-2pi k/2)/2))

120/pi k(cos(2 pi k/2)sin (-pi k/2) )

120/pi k (cos (pi k)(-sin(pi k/2)))

And cos pi k =-1 for odd values of k

120/pi k (-1*- sin (pi k/2))

120/pi k * sin (pi k)/2