Let, P(n) be the given statement,
now, P(n); sin x + sin 3x + ....+ sin(2n-1)x = \(\frac{sin^2nx}{sin\,x}\)
step1: P(n): sin x = \(\frac{sin^2nx}{sin\,x}\)
Thus, P(1) is true.
Step2: Let, P(m) be true.
then, sin x + sin 3x + ....+ sin(2m-1)x = \(\frac{sin^2mx}{sin\,x}\)
Now, we need to show that P(m+1) is true when P(m) is true.
As P(m) is true
Thus, P(m+1) is divisible by x+y. So, by the principle of mathematical induction P(n) is true for all n.