Question

$\mathrm{Prove}\mathrm{that}\cos \alpha +\cos \left(\alpha +\beta \right)+\cos \left(\alpha +2\beta \right)+...+\cos \left[\alpha +\left(n-1\right)\beta \right]=\frac{\cos \left\{\alpha +\left({\displaystyle \frac{n-1}{2}}\right)\beta \right\}\sin \left({\displaystyle \frac{n\beta }{2}}\right)}{\sin \left({\displaystyle \frac{\beta }{2}}\right)}\mathrm{for}\mathrm{all}n\in \mathbf{N}.$ [NCERT EXEMPLAR]

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{p}\left(n\right):\cos \alpha +\cos \left(\alpha +\beta \right)+\cos \left(\alpha +2\beta \right)+...+\cos \left[\alpha +\left(n-1\right)\beta \right]=\frac{\cos \left\{\alpha +\left({\displaystyle \frac{n-1}{2}}\right)\beta \right\}\sin \left({\displaystyle \frac{n\beta }{2}}\right)}{\sin \left({\displaystyle \frac{\beta }{2}}\right)}\forall n\in \mathbf{N}. \\ \mathrm{Step}\mathrm{I}:\mathrm{For}n=1, \\ \mathrm{LHS}=\cos \left[\alpha +\left(1-1\right)\beta \right]=\cos \alpha \\ \mathrm{RHS}=\frac{\cos \left\{\alpha +\left({\displaystyle \frac{1-1}{2}}\right)\beta \right\}\sin \left({\displaystyle \frac{\beta }{2}}\right)}{\sin \left({\displaystyle \frac{\beta }{2}}\right)}=\cos \alpha \\ \mathrm{As},\mathrm{LHS}=\mathrm{RHS} \\ \mathrm{So},\mathrm{it}\mathrm{is}\mathrm{true}\mathrm{for}n=1. \\ \mathrm{Step}\mathrm{II}:\mathrm{For}n=k, \\ \mathrm{Let}\mathrm{p}\left(k\right):\cos \alpha +\cos \left(\alpha +\beta \right)+\cos \left(\alpha +2\beta \right)+...+\cos \left[\alpha +\left(k-1\right)\beta \right]=\frac{\cos \left\{\alpha +\left({\displaystyle \frac{k-1}{2}}\right)\beta \right\}\sin \left({\displaystyle \frac{k\beta }{2}}\right)}{\sin \left({\displaystyle \frac{\beta }{2}}\right)}\mathrm{be}\mathrm{true}\forall k\in \mathbf{N}. \\ \mathrm{Step}\mathrm{III}:\mathrm{For}n=k+1, \\ \mathrm{LHS}=\cos \alpha +\cos \left(\alpha +\beta \right)+\cos \left(\alpha +2\beta \right)+...+\cos \left[\alpha +\left(k-1\right)\beta \right]+\cos \left[\alpha +\left(k+1-1\right)\beta \right] \\ =\frac{\cos \left\{\alpha +\left({\displaystyle \frac{k-1}{2}}\right)\beta \right\}\sin \left({\displaystyle \frac{k\beta }{2}}\right)}{\sin \left({\displaystyle \frac{\beta }{2}}\right)}+\cos \left(\alpha +k\beta \right) \\ =\frac{\cos \left\{\alpha +\left({\displaystyle \frac{k-1}{2}}\right)\beta \right\}\sin \left({\displaystyle \frac{k\beta }{2}}\right)+\sin \left({\displaystyle \frac{\beta }{2}}\right)\cos \left(\alpha +k\beta \right)}{\sin \left({\displaystyle \frac{\beta }{2}}\right)} \\ =\frac{\sin \left(\alpha +k\beta -{\displaystyle \frac{\beta }{2}}\right)-\sin \left(\alpha -{\displaystyle \frac{\beta }{2}}\right)+\sin \left({\displaystyle \alpha +k\beta +\frac{\mathrm{\beta }}{2}}\right)-\sin \left(\alpha +k\beta -{\displaystyle \frac{\beta }{2}}\right)}{2\sin \left({\displaystyle \frac{\mathrm{\beta }}{2}}\right)} \\ =\frac{-\sin \left(\alpha -{\displaystyle \frac{\beta }{2}}\right)+\sin \left({\displaystyle \alpha +k\beta +\frac{\beta }{2}}\right)}{2\sin \left({\displaystyle \frac{\mathrm{\beta }}{2}}\right)} \\ =\frac{2\cos \left({\displaystyle \frac{2\alpha +k\beta }{2}}\right)\sin \left({\displaystyle \frac{k\beta +\beta }{2}}\right)}{2\sin \left({\displaystyle \frac{\beta }{2}}\right)} \\ =\frac{\cos \left({\displaystyle \alpha +\frac{{\displaystyle k\beta }}{2}}\right)\sin \left({\displaystyle \frac{\left(k+1\right)\beta }{2}}\right)}{\sin \left({\displaystyle \frac{\beta }{2}}\right)} \\ \mathrm{RHS}=\frac{\cos \left\{\alpha +\left({\displaystyle \frac{k+1-1}{2}}\right)\beta \right\}\sin \left({\displaystyle \frac{\left(k+1\right)\beta }{2}}\right)}{\sin \left({\displaystyle \frac{\beta }{2}}\right)} \\ =\frac{\cos \left({\displaystyle \alpha +\frac{{\displaystyle k\beta }}{2}}\right)\sin \left({\displaystyle \frac{\left(k+1\right)\beta }{2}}\right)}{\sin \left({\displaystyle \frac{\beta }{2}}\right)} \\ \mathrm{As},\mathrm{LHS}=\mathrm{RHS} \\ \mathrm{So},\mathrm{it}\mathrm{is}\mathrm{also}\mathrm{true}\mathrm{for}n=k+1. \end{gathered}$$

$\mathrm{Hence},\cos \alpha +\cos \left(\alpha +\beta \right)+\cos \left(\alpha +2\beta \right)+...+\cos \left[\alpha +\left(n-1\right)\beta \right]=\frac{\cos \left\{\alpha +\left({\displaystyle \frac{n-1}{2}}\right)\beta \right\}\sin \left({\displaystyle \frac{n\beta }{2}}\right)}{\sin \left({\displaystyle \frac{\beta }{2}}\right)}\mathrm{for}\mathrm{all}n\in \mathbf{N}.$