Expand cos²x in a series of powers of (x- dfrac pi4)
Expand $\cos^2x$ in a series of powers of $\left(x-\dfrac{\pi}4\right)$.
Explanation of Method
The identity $\cos^2x=\frac{1+\cos2x}2$ turns this into an expansion of $\cos(2x)$ shifted so that $2x=\frac\pi2+2y$ with $y=x-\pi/4$; since $\cos(\theta+\pi/2)=-\sin\theta$, the problem reduces to expanding the ordinary sine series in the variable $2y$.
- Use the double-angle identity and shift to $y=x-\pi/4$.
- Simplify $\cos(2y+\pi/2)=-\sin(2y)$ and expand $\sin(2y)$ via its Maclaurin series.
- Assemble the final series and note it converges for all $x$.
Step 1 · Shift and simplify
With $y=x-\pi/4$: $\cos^2x=\dfrac{1+\cos2x}2$ and $2x=2y+\dfrac\pi2$, so using $\cos(\theta+\pi/2)=-\sin\theta$: $$ \cos^2x = \frac{1+\cos\left(2y+\frac\pi2\right)}2 = \frac{1-\sin2y}2. $$
Step 2 · Expand $\sin(2y)$
$$ \sin2y = \sum_{k=0}^{\infty}\frac{(-1)^k(2y)^{2k+1}}{(2k+1)!} = 2y-\frac{4y^3}3+\frac{4y^5}{15}-\cdots $$
Step 3 · Assemble the final series
$$ \cos^2x = \frac12-\frac12\sin2y = \frac12-\sum_{k=0}^{\infty}\frac{(-1)^k2^{2k}}{(2k+1)!}y^{2k+1} = \frac12-y+\frac23y^3-\frac2{15}y^5+\cdots $$ Substituting back $y=x-\pi/4$: $$ \cos^2x = \frac12-\left(x-\frac\pi4\right)+\frac23\left(x-\frac\pi4\right)^3-\frac2{15}\left(x-\frac\pi4\right)^5+\cdots $$ Since $\sin(2y)$ is entire, this series converges for every real $x$.
Summary Table
| Property | Value |
|---|---|
| General term coefficient | $\dfrac{(-1)^{k+1}2^{2k}}{(2k+1)!}$ |
| $\cos^2(\pi/4)$ | $1/2$ |
| Region of validity | $\mathbb R$ |
Additional Notes
- Only odd powers of $\left(x-\frac\pi4\right)$ appear, since $\cos^2x-\frac12=-\frac12\sin(2y)$ is an odd function of $y=x-\pi/4$ — the point $x=\pi/4$ is exactly the inflection-symmetric center of the shifted sine wave.
- As a composition of entire functions, $\cos^2x$ inherits the infinite radius of convergence from $\sin$ and $\cos$ directly — the same conclusion reached for $\sinh x$, $\cosh x$, and $a^x$ in Problems 252–254.
- Equivalent derivative-based computation confirms the same coefficients: even-order derivatives of $\cos^2x$ vanish at $x=\pi/4$ (except the constant term), while odd-order derivatives alternate in sign, matching this series' odd-powers-only structure exactly.