sinx+sin2x+sin3x+...+sin nx=sin((n+1)/2)x .((sin nx)/2) / sin(x/2)
step I : for n = 1
thus the result is true for n = 1.
Step II: let it is true for n = k.
now in the third step , we will try to prove that it is true for n = (k+1).
thus it is true for n = (k+1). therefore it will be true for all values of n belongs to N.
hope this helps you.