I am having trouble proving the third part of the problem. "Show if $0 < \theta < 2\pi $, $\left| \sum _{n=1}^{p}\sin \left( n\theta \right) \right| < \cos ec\dfrac {\theta } {2}$; and deduce that, if $f_{n}\rightarrow 0$ steadily(synonym for monotonically), $\sum _{n=1}^{\infty }f_{n}\sin \left( n\theta \right) $ converges for all real values of $\theta$, and that $\sum _{n=1}^{\infty }f_{n}\cos \left( n\theta \right) $ converges if $\theta$ is not even multiple of $\pi$.
Here is where i got upto, to see that the inequality holds i used Lagrange's trigonometric identity in the LHS and the result follows. Once that is assumed to hold, $\sum _{n=1}^{\infty }f_{n}\sin \left( n\theta \right) $ just follows from Dirichlet's test of convergence.
I am unsure how to show the last one i assume the problem is some how related to Abel's inequality or Dirichlet's test of convergence, any help would be much appreciated.