For each $n \in N$,which of the following statements is correct?

Use the Principle of Mathematical Induction to prove that for all $n \in N$:
$\sin \theta + \sin 2\theta + \ldots + \sin n\theta = \frac{\sin \frac{n\theta}{2} \sin \frac{(n+1)\theta}{2}}{\sin \frac{\theta}{2}}$

Prove the following by using the principle of mathematical induction for all $n \in N$:
$1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n(n+1) = \frac{n(n+1)(n+2)}{3}$

Prove that for all $n \in N$,$3^{2n+2} - 8n - 9$ is divisible by $8$ using the principle of mathematical induction.

Let $P(n): 1^2+2^2+3^2+\ldots+n^2 = \frac{6(n-1)(n-2) \ldots(n-2020)+2n^3+3n^2+n}{6}$,for all $n \in N$. Then which of the following is correct?

Prove the statement by the Principle of Mathematical Induction: For any natural number $n$,$x^{n}-y^{n}$ is divisible by $x-y$,where $x$ and $y$ are any integers with $x \neq y$.