Question

Find the sum of $1+|\cos \theta |+{\cos }^2\theta +|{\cos }^3\theta |+....cos(n-1)\theta$

Answer 1

$\begin{array}{c}M=\cos \theta +\cos 2\theta +\cos 3\theta +..........+\cos \left(n-1\right)\theta \\ M=\frac{2\sin \frac{\theta }{2}}{2\sin \frac{\theta }{2}}\left(\cos \theta +\cos 2\theta +..........\right)\\ =\frac{\left(\sin \frac{3\theta }{2}-\sin \frac{\theta }{2}\right)+\left(\sin \frac{5\theta }{2}-\sin \frac{3\theta }{2}\right)+........}{2\sin \frac{\theta }{2}}\\ M=\frac{-\sin \frac{\theta }{2}+\sin \left(\frac{n+1}{2}\right)\theta }{2\sin \frac{\theta }{2}}\\ \therefore M+1=1+\frac{-\sin \frac{\theta }{2}+\sin \left(\frac{n+1}{2}\right)\theta }{2\sin \frac{\theta }{2}}\\ M+1=\frac{\sin \frac{\theta }{2}+\sin \left(\frac{n+1}{2}\right)\theta }{2\sin \frac{\theta }{2}}\end{array}$