Use the Principle of Mathematical Induction to prove that for all $n \in N$:
$\cos \alpha + \cos (\alpha + \beta) + \cos (\alpha + 2\beta) + \ldots + \cos [\alpha + (n-1)\beta] = \frac{\cos \left[\alpha + \left(\frac{n-1}{2}\right) \beta\right] \sin \left(\frac{n\beta}{2}\right)}{\sin \left(\frac{\beta}{2}\right)}$

(A) Let $P(n): \cos \alpha + \cos (\alpha + \beta) + \ldots + \cos [\alpha + (n-1)\beta] = \frac{\cos \left[\alpha + \left(\frac{n-1}{2}\right) \beta\right] \sin \left(\frac{n\beta}{2}\right)}{\sin \left(\frac{\beta}{2}\right)}$
For $n=1$,$L.H.S. = \cos \alpha$
$R.H.S. = \frac{\cos \left[\alpha + \left(\frac{1-1}{2}\right) \beta\right] \sin \left(\frac{\beta}{2}\right)}{\sin \left(\frac{\beta}{2}\right)} = \cos \alpha$
Since $L.H.S. = R.H.S.$,$P(1)$ is true.
Assume $P(k)$ is true for some $k \in N$:
$P(k): \cos \alpha + \cos (\alpha + \beta) + \ldots + \cos [\alpha + (k-1)\beta] = \frac{\cos \left[\alpha + \left(\frac{k-1}{2}\right) \beta\right] \sin \left(\frac{k\beta}{2}\right)}{\sin \left(\frac{\beta}{2}\right)} \quad \ldots (i)$
For $n=k+1$,we need to prove:
$P(k+1): \sum_{i=0}^{k} \cos(\alpha + i\beta) = \frac{\cos \left[\alpha + \frac{k\beta}{2}\right] \sin \left(\frac{(k+1)\beta}{2}\right)}{\sin \left(\frac{\beta}{2}\right)}$
$L.H.S. = [\cos \alpha + \ldots + \cos(\alpha + (k-1)\beta)] + \cos(\alpha + k\beta)$
$= \frac{\cos \left[\alpha + \frac{(k-1)\beta}{2}\right] \sin \left(\frac{k\beta}{2}\right)}{\sin \left(\frac{\beta}{2}\right)} + \cos(\alpha + k\beta)$
$= \frac{\cos \left(\alpha + \frac{k\beta}{2} - \frac{\beta}{2}\right) \sin \left(\frac{k\beta}{2}\right) + \sin \left(\frac{\beta}{2}\right) \cos(\alpha + k\beta)}{\sin \left(\frac{\beta}{2}\right)}$
Using $\cos(A-B) = \cos A \cos B + \sin A \sin B$ and $\cos(A+B) = \cos A \cos B - \sin A \sin B$:
$= \frac{[\cos(\alpha + \frac{k\beta}{2}) \cos \frac{\beta}{2} + \sin(\alpha + \frac{k\beta}{2}) \sin \frac{\beta}{2}] \sin \frac{k\beta}{2} + \sin \frac{\beta}{2} \cos(\alpha + k\beta)}{\sin \frac{\beta}{2}}$
After simplification using trigonometric identities,we get:
$= \frac{\cos(\alpha + \frac{k\beta}{2}) [\sin \frac{k\beta}{2} \cos \frac{\beta}{2} + \cos \frac{k\beta}{2} \sin \frac{\beta}{2}]}{\sin \frac{\beta}{2}}$
$= \frac{\cos(\alpha + \frac{k\beta}{2}) \sin(\frac{k\beta + \beta}{2})}{\sin \frac{\beta}{2}} = \frac{\cos(\alpha + \frac{k\beta}{2}) \sin(\frac{(k+1)\beta}{2})}{\sin \frac{\beta}{2}}$
Thus,$P(k+1)$ is true. By the Principle of Mathematical Induction,$P(n)$ is true for all $n \in N$.