From the result of the last section and the equations (1) of § 228 it follows at once that $\cos z = 1 – \frac{z^{2}}{2!} + \frac{z^{4}}{4!} – \dots,\quad \sin z = z – \frac{z^{3}}{3!} + \frac{z^{5}}{5!} – \dots$ for all values of $$z$$. These results were proved for real values of $$z$$ in Ex. LVI. 1.

Example XCVI
1. Calculate $$\cos i$$ and $$\sin i$$ to two places of decimals by means of the power series for $$\cos z$$ and $$\sin z$$.

2. Prove that $$|\cos z| \leq \cosh|z|$$ and $$|\sin z| \leq \sinh|z|$$.

3. Prove that if $$|z| < 1$$ then $$|\cos z| < 2$$ and $$|\sin z| < \frac{6}{5}|z|$$.

4. Since $$\sin 2z = 2\sin z \cos z$$ we have $(2z) – \frac{(2z)^{3}}{3!} + \frac{(2z)^{5}}{5!} – \dots = 2\left(z – \frac{z^{3}}{3!} + \dots\right) \left(1 – \frac{z^{2}}{2!} + \dots\right).$ Prove by multiplying the two series on the right-hand side (§ 195) and equating coefficients (§ 194) that $\binom{2n + 1}{1} + \binom{2n + 1}{3} + \dots + \binom{2n + 1}{2n + 1} = 2^{2n}.$ Verify the result by means of the binomial theorem. Derive similar identities from the equations $\cos^{2}z + \sin^{2}z = 1,\quad \cos2z = 2\cos^{2}z – 1 = 1 – 2\sin^{2}z.$

5. Show that $\exp\{(1 + i)z\} = \sum_{0}^{\infty} 2^{\frac{1}{2}n} \exp(\tfrac{1}{4}n\pi i) \frac{z^{n}}{n!}.$

6. Expand $$\cos z \cosh z$$ in powers of $$z$$. [We have \begin{aligned} \cos z \cosh z + i\sin z \sinh z &= \cos\{(1 – i)z\} = \tfrac{1}{2} [\exp\{(1 + i)z\} + \exp\{-(1 + i)z\}]\\ &= \tfrac{1}{2} \sum_{0}^{\infty} 2^{\frac{1}{2}n} \{1 + (-1)^{n}\} \exp(\tfrac{1}{4}n\pi i) \frac{z^{n}}{n!},\end{aligned} and similarly $\cos z \cosh z – i\sin z \sinh z = \cos (1 + i)z = \tfrac{1}{2} \sum_{0}^{\infty} 2^{\frac{1}{2}n} \{1 + (-1)^{n}\} \exp(-\tfrac{1}{4}n\pi i) \frac{z^{n}}{n!}.$ Hence $\cos z \cosh z = \tfrac{1}{2} \sum_{0}^{\infty} 2^{\frac{1}{2}n}\{1 + (-1)^{n}\} \cos \tfrac{1}{4}n\pi \frac{z^{n}}{n!} = 1 – \frac{2^{2}z^{4}}{4!} + \frac{2^{4}z^{8}}{8!} – \dots.]$

7. Expand $$\sin z \sinh z$$, $$\cos z \sinh z$$, and $$\sin z \cosh z$$ in powers of $$z$$.

8. Expand $$\sin^{2} z$$ and $$\sin^{3} z$$ in powers of $$z$$. [Use the formulae $\sin^{2} z = \tfrac{1}{2} (1 – \cos 2z),\quad \sin^{3} z = \tfrac{1}{4} (3\sin z – \sin 3z),\ \dots.$ It is clear that the same method may be used to expand $$\cos^{n} z$$ and $$\sin^{n} z$$, where $$n$$ is any integer.]

9. Sum the series $C = 1 + \frac{\cos z}{1!} + \frac{\cos 2z}{2!} + \frac{\cos 3z}{3!} +\dots,\quad S = \frac{\sin z}{1!} + \frac{\sin 2z}{2!} + \frac{\sin 3z}{3!} + \dots.$

[Here \begin{aligned} C + iS &= 1 + \dfrac{\exp(iz)}{1!} + \dfrac{\exp(2iz)}{2!} + \dots = \exp\{\exp(iz)\} \\ &= \exp(\cos z) \{\cos(\sin z) + i\sin(\sin z)\},\end{aligned} and similarly $C – iS = \exp\{\exp(-iz)\} = \exp(\cos z)\{\cos(\sin z) – i\sin(\sin z)\}.$ Hence $C = \exp(\cos z)\cos(\sin z),\quad S = \exp(\cos z)\sin(\sin z).]$

10. Sum $1 + \frac{a\cos z}{1!} + \frac{a^{2}\cos 2z}{2!} + \dots,\quad \frac{a\sin z}{1!} + \frac{a^{2}\sin 2z}{2!} + \dots.$

11. Sum $1 – \frac{\cos 2z}{2!} + \frac{\cos 4z}{4!} – \dots,\quad \frac{\cos z}{1!} – \frac{\cos 3z}{3!} + \dots$ and the corresponding series involving sines.

12. Show that $1 + \frac{\cos 4z}{4!} + \frac{\cos 8z}{8!} + \dots = \tfrac{1}{2}\{\cos(\cos z) \cosh(\sin z) + \cos(\sin z) \cosh(\cos z)\}.$

13. Show that the expansions of $$\cos(x + h)$$ and $$\sin(x + h)$$ in powers of $$h$$ (Ex. LVI. 1) are valid for all values of $$x$$ and $$h$$, real or complex.