Chapter 10: Infinite Sequences and Series - Section 10.10 - The Binomial Series and Applications of Taylor Series - Exercises 10.10 - Page 633: 43

$\cos (\dfrac{3}{4})$

As we know that $\cos x=1-\dfrac{x^2}{2!}-.....+(-1)^n \dfrac{x^{2n}}{(2n)!}$ Now, we have $1-(\dfrac{3^2}{4^2 (2!)})+(\dfrac{3^4}{4^4 (4!)})-(\dfrac{3^6}{4^6 (6!)})+....=1-(\dfrac{1}{2!})(\dfrac{3}{4})^2+(\dfrac{1}{4!})(\dfrac{3}{4})^4-...$ This implies that $1-(\dfrac{1}{2!})(\dfrac{3}{4})^2+(\dfrac{1}{4!})(\dfrac{3}{4})^4-...=\cos (\dfrac{3}{4})$