If ${\alpha }_n=\underset{0}{\overset{\frac{I}{2}}{\int }}{\cos }^{n}x\cos nxdxand{\beta }_n=\underset{0}{\overset{I}{\int }}\frac{\sin \left(nx\right)}{\sin x}dx$  then for the sequence $\left\{{\alpha }_n\right\}and\left\{{\beta }_n\right\}$  : {where I = $K\pi ,K\in N$ } 

$$\begin{gathered} {\alpha }_{n-1}=\underset{0}{\overset{\frac{I}{2}}{\int }}{\cos }^{n-1}x\cos (n-1)xdx \\ {\alpha }_{n-1}=\underset{0}{\overset{\frac{I}{2}}{\int }}{\cos }^{n-1}x(\cos nx.\cos x+\sin nx.\sin x)dx \\ {\alpha }_{n-1}=\underset{0}{\overset{I/2}{\int }}{\cos }^{n}x.\cos nxdx+\underset{0}{\overset{I/2}{\int }}\frac{{\cos }^{n-1}x\sin x}{IBP}\sin \left(nx\right)dx \\ {\alpha }_{n-1}={\alpha }_n+{\alpha }_n\Rightarrow {\alpha }_{n-1}=2{\alpha }_n \\ {\beta }_n=\{\begin{array}{cc}0,& ifn\in even\\ I,& ifn\in odd\end{array} \end{gathered}$$