Compute Taylor error of $10^{-5}$ for $\cos(\pi/20)$
Solution 1:
No, it is not correct. You wrote $R_2(X)$, but I suppose that you meant $R_n(X)$. Furthermore,$$\left|R_n\left(\frac\pi{20}\right)\right|\leqslant\frac{\left(\frac\pi{20}\right)^{n+1}}{(n+1)!}.$$Using the fact that $\pi<4$ (and that therefore $\frac\pi{20}<\frac15$), you can see that taking $n=4$ is enough.