Q.Prove using the Mathematical induction

[!FORMULA] $\sin(\alpha)+\sin\left(\alpha+\dfrac{\pi}{6}\right)+\sin\left(\alpha+\dfrac{2\pi}{6}\right)+\cdots+\sin\left(\alpha+\dfrac{(n-1)\pi}{6}\right) = \dfrac{\sin\left(\alpha+\dfrac{(n-1)\pi}{12}\right)\times\sin\left(\dfrac{n\pi}{12}\right)}{\sin\left(\dfrac{\pi}{12}\right)}.$

The Principle of Mathematical Induction (PMI) is a method for proving that a statement $P(n)$, framed for every natural number $n$, is true for all $n \in \mathbb{N}$ at once — without needing to verify infinitely many individual cases directly. It rests on two pillars: a base case, where $P(1)$ (or the smallest relevant value of $n$) is checked directly, and an inductive step, where it is shown that the truth of $P(k)$ forces the truth of $P(k+1)$ for every positive integer $k$. Once both are established, PMI concludes that $P(n)$ holds for every natural number, be …