Prove that: `\ sin x+sin3x++sin(2n-1)x=(sin^2\ \ n x)/(sin x)` for all `n in NN`

Explanation

To prove the statement \( S(n): \sin x + \sin 3x + \sin (2n-1)x = \frac{\sin^2(nx)}{\sin x} \) for all \( n \in \mathbb{N} \), we will use the principle of mathematical induction.

### Step 1: Base Case
We first check the base case for \( n = 1 \).

**Left Hand Side (LHS)**:
\[
S(1) = \sin x
...