Proof of Continuity, Derivatives and Power Series of Sine and Cosine

Routine facts are used without comment (Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity), and the earlier and the new versions of the definitions involved are read through The Earlier and the New Versions of the Definitions of the Natural Numbers, the Complex Numbers, the Conjugate and the Modulus Agree, and the Two Versions of Real Correspond (ii), The Earlier and the New Versions of the Definitions of the Natural Numbers, the Complex Numbers, the Conjugate and the Modulus Agree, and the Two Versions of Real Correspond (iv), The Earlier and the New Versions of the Definitions of the Derivative on an Interval and of the Riemann Integral Agree (ii), The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (i), The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (ii), The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (iii) and The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (v). Throughout, $\exp:\mathbb{C}\to\mathbb{C}$ is the complex exponential function, so that $\cos s=\operatorname{Re}\exp(is)$ and $\sin s=\operatorname{Im}\exp(is)$ for every $s\in\mathbb{R}$ (The Real Sine and Cosine Functions (i), The Real Sine and Cosine Functions (ii)).

Two facts about the modulus. As $i=0+1\cdot i$, $|i|=\sqrt{0^{2}+1^{2}}=1$ by The Modulus of a Complex Number (i), and on real numbers the modulus is the absolute value (The Modulus of a Complex Number (ii)); so multiplicativity (Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (iv)) gives, for every $v\in\mathbb{C}$ and $s\in\mathbb{R}$,

$$|iv|=|v|,\qquad |is|=|s|. \tag{1}$$

Derivatives. Fix $x\in\mathbb{R}$. By Euler's formula (Euler's Formula: the Complex Exponential on the Imaginary Axis, Its Modulus and Its Conjugate (i)),

$$i\exp(ix)=i(\cos x+i\sin x)=-\sin x+i\cos x, \tag{2}$$

using $i^{2}=-1$ (The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers (vii)).

Let $\varepsilon>0$ be given; then choose

$$\delta=\min\Bigl(1,\frac{\varepsilon}{e}\Bigr),$$

which is positive because $e=\exp_{\mathbb{R}}(1)>0$ (Complex Analysis: Standing Notation (iv), Basic Properties of the Exponential Function (iii)). Let $u\in\mathbb{R}$ with $0<|u-x|<\delta$, put $h=u-x\neq0$, and set

$$w=\frac{\exp(iu)-\exp(ix)}{h}-i\exp(ix)=i\left[\frac{\exp(ix+ih)-\exp(ix)}{ih}-\exp(ix)\right],$$

where $iu=ix+ih$ and $ih\neq0$. The difference-quotient estimate for the exponential (Estimates for the Complex Exponential and Its Continuity (iii)), applied at the point $ix$ with the increment $ih$, together with (1), gives

$$|w|\le|\exp(ix)|\,|ih|\,\exp_{\mathbb{R}}(|ih|)=|h|\,\exp_{\mathbb{R}}(|h|),$$

since $|\exp(ix)|=1$ (Euler's Formula: the Complex Exponential on the Imaginary Axis, Its Modulus and Its Conjugate (ii)). As $|h|<1$, the real exponential being increasing (Basic Properties of the Exponential Function (ix)) gives $\exp_{\mathbb{R}}(|h|)<\exp_{\mathbb{R}}(1)=e$, and therefore

$$|w|<e\,|h|<e\cdot\frac{\varepsilon}{e}=\varepsilon .$$

Now take real and imaginary parts of $w$. For the real number $h\neq0$ and $v\in\mathbb{C}$, $\operatorname{Re}(v/h)=(\operatorname{Re}v)/h$ and $\operatorname{Im}(v/h)=(\operatorname{Im}v)/h$, and real and imaginary parts are additive (Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (ii)). Applying this to $v=\exp(iu)-\exp(ix)$ and using (2),

$$\operatorname{Im}w=\frac{\sin u-\sin x}{u-x}-\cos x,\qquad \operatorname{Re}w=\frac{\cos u-\cos x}{u-x}+\sin x .$$

Both are bounded in absolute value by $|w|$ (Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (vi)), so

$$\Bigl|\frac{\sin u-\sin x}{u-x}-\cos x\Bigr|<\varepsilon,\qquad \Bigl|\frac{\cos u-\cos x}{u-x}-(-\sin x)\Bigr|<\varepsilon$$

for every $u\in\mathbb{R}$ with $0<|u-x|<\delta$. Thus $\cos x$ is a derivative of $\sin$ at $x$ and $-\sin x$ is a derivative of $\cos$ at $x$ (Derivative of a Real Function on an Interval (i)); so $\sin$ and $\cos$ are differentiable at $x$ with $\sin'(x)=\cos x$ and $\cos'(x)=-\sin x$ (Derivative of a Real Function on an Interval (ii)). As $x$ was arbitrary, $\sin$ and $\cos$ are differentiable on $\mathbb{R}$ (Derivative of a Real Function on an Interval (iii)), and their derivatives $\sin',\cos':\mathbb{R}\to\mathbb{R}$ take the values stated in (ii) (Derivative of a Real Function on an Interval (iv)).

Continuity. By (ii), $\sin$ and $\cos$ are differentiable at every $x\in\mathbb{R}$, so each is continuous at every $x$ relative to $\mathbb{R}$, because differentiability implies continuity (The Increment Form of the Derivative, and Differentiability Implies Continuity (ii)). By the characterization of continuity on a set in The Real Line: Standing Notation and Background for Calculus on Intervals (vi), $\sin$ and $\cos$ are continuous on $\mathbb{R}$. This proves (i).

Power series. Fix $t\in\mathbb{R}$. Let $(a_{n})$ be the sequence in $\mathbb{C}$ with

$$a_{n}=\frac{(it)^{n-1}}{(n-1)!}\qquad(n\in\mathbb{N})$$

(The Complex Exponential Series Converges Absolutely, and Its Sum Is a Function on ℂ (i)), and let $s_{n}=\sum_{j=1}^{n}a_{j}$ be its partial sums (Series of Complex Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (i)). For $k\in\mathbb{N}$ write

$$c_{k}=\frac{(-1)^{k-1}t^{2k-2}}{(2k-2)!},\qquad \sigma_{k}=\frac{(-1)^{k-1}t^{2k-1}}{(2k-1)!},$$

and let $C_{m}=\sum_{k=1}^{m}c_{k}$ and $S_{m}=\sum_{k=1}^{m}\sigma_{k}$ be the partial sums of these two real sequences (Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (i)).

Since $i^{2}=-1$ (The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers (vii)), we have $(it)^{2k-2}=(i^{2})^{k-1}t^{2k-2}=(-1)^{k-1}t^{2k-2}$ and $(it)^{2k-1}=i\,(-1)^{k-1}t^{2k-1}$, so

$$a_{2k-1}=c_{k},\qquad a_{2k}=i\sigma_{k}\qquad(k\in\mathbb{N}). \tag{3}$$

We claim that

$$s_{2m}=C_{m}+iS_{m}\qquad(m\in\mathbb{N}). \tag{4}$$

For $m=1$, $s_{2}=a_{1}+a_{2}=1+it=C_{1}+iS_{1}$. If (4) holds for $m$, then by (3)

$$s_{2m+2}=s_{2m}+a_{2m+1}+a_{2m+2}=(C_{m}+c_{m+1})+i(S_{m}+\sigma_{m+1})=C_{m+1}+iS_{m+1},$$

so (4) holds for all $m$ by induction. As $C_{m}$ and $S_{m}$ are real, (4) gives

$$\operatorname{Re}s_{2m}=C_{m},\qquad \operatorname{Im}s_{2m}=S_{m} \tag{5}$$

(Real and Imaginary Parts of a Complex Number (i)).

The exponential series at $it$ converges (The Complex Exponential Series Converges Absolutely, and Its Sum Is a Function on ℂ (ii)), and its sum is $\exp(it)$ (The Complex Exponential Function (i)), so $s_{n}\to\exp(it)$ (Series of Complex Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (ii)). Taking real and imaginary parts (Limits of Complex Sequences: Uniqueness, Real and Imaginary Parts, Arithmetic, Real Sequences, Conjugates, Moduli, the Cauchy Criterion and the Modulus Metric (ii)),

$$\operatorname{Re}s_{n}\to\operatorname{Re}\exp(it)=\cos t,\qquad \operatorname{Im}s_{n}\to\operatorname{Im}\exp(it)=\sin t .$$

The map $m\mapsto2m$ from $\mathbb{N}$ to $\mathbb{N}$ is strictly increasing, so by (5) the sequences $(C_{m})$ and $(S_{m})$ are subsequences of $(\operatorname{Re}s_{n})$ and $(\operatorname{Im}s_{n})$. Every subsequence of a convergent sequence converges to the same limit (Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (xiii)), hence $C_{m}\to\cos t$ and $S_{m}\to\sin t$. That is, the two series of (iii) converge, with sums $\cos t$ and $\sin t$ (Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (ii)).

It remains to show absolute convergence. By (3), (1) and The Modulus of a Complex Number (ii), $|c_{k}|=|a_{2k-1}|$ and $|\sigma_{k}|=|i\sigma_{k}|=|a_{2k}|$. The series $\sum_{n=1}^{\infty}|a_{n}|$ converges (The Complex Exponential Series Converges Absolutely, and Its Sum Is a Function on ℂ (ii), Series of Complex Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (iii)); call its sum $A$. Its terms are nonnegative, so each of its partial sums is at most $A$ (Series of Real Numbers with Nonnegative Terms, Comparison, Absolute Convergence and the Geometric Series (i)). Hence, for every $m\in\mathbb{N}$,

$$\sum_{k=1}^{m}|c_{k}|+\sum_{k=1}^{m}|\sigma_{k}|=\sum_{k=1}^{m}\bigl(|a_{2k-1}|+|a_{2k}|\bigr)=\sum_{n=1}^{2m}|a_{n}|\le A .$$

So the partial sums of $\sum_{k=1}^{\infty}|c_{k}|$ and of $\sum_{k=1}^{\infty}|\sigma_{k}|$ are bounded above by $A$, and since these series have nonnegative terms they converge (Series of Real Numbers with Nonnegative Terms, Comparison, Absolute Convergence and the Geometric Series (i)). That is, both series of (iii) converge absolutely (Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (iii)), which completes the proof.