Produce a power series for cos^2 2 x as far as the term in...

Produce a power series for $\cos ^{2} 2 x$ as far as the term in $x^{6}$ From double angle formulae, $\cos 2 A=2 \cos ^{2} A-1$ (see Chapter 19). from which, $\quad \cos ^{2} A=\frac{1}{2}(1+\cos 2 A)$ and $\quad \cos ^{2} 2 x=\frac{1}{2}(1+\cos 4 x)$ From Problem 1 , $$ \begin{aligned} &\cos x=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\frac{x^{6}}{6 !}+\cdots \\ &\text { hence } \quad \cos 4 x=1-\frac{(4 x)^{2}}{2 !}+\frac{(4 x)^{4}}{4 !}-\frac{(4 x)^{6}}{6 !}+\cdots \\ &=1-8 x^{2}+\frac{32}{3} x^{4}-\frac{256}{45} x^{6}+\cdots \end{aligned} $$ $$ =\frac{1}{2}\left(1+1-8 x^{2}+\frac{32}{3} x^{4}-\frac{256}{45} x^{6}+\cdots\right) $$ i.e. $\cos ^{2} 2 x=1-4 x^{2}+\frac{16}{3} x^{4}-\frac{128}{45} x^{6}+\cdots$

Produce a power series for $\cos ^{2} 2 x$ as far as the term in $x^{6}$
From double angle formulae, $\cos 2 A=2 \cos ^{2} A-1$ (see Chapter 19).
from which, $\quad \cos ^{2} A=\frac{1}{2}(1+\cos 2 A)$ and $\quad \cos ^{2} 2 x=\frac{1}{2}(1+\cos 4 x)$
From Problem 1 ,
$$
\begin{aligned}
&\cos x=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\frac{x^{6}}{6 !}+\cdots \\
&\text { hence } \quad \cos 4 x=1-\frac{(4 x)^{2}}{2 !}+\frac{(4 x)^{4}}{4 !}-\frac{(4 x)^{6}}{6 !}+\cdots \\
&=1-8 x^{2}+\frac{32}{3} x^{4}-\frac{256}{45} x^{6}+\cdots
\end{aligned}
$$
$$
=\frac{1}{2}\left(1+1-8 x^{2}+\frac{32}{3} x^{4}-\frac{256}{45} x^{6}+\cdots\right)
$$
i.e. $\cos ^{2} 2 x=1-4 x^{2}+\frac{16}{3} x^{4}-\frac{128}{45} x^{6}+\cdots$

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Key Concepts

Power Series Expansion

A power series expansion expresses a function as an infinite sum of powers of the variable, typically centered around a specific point such as 0. This concept is important for approximating functions and analyzing their behavior term by term, as seen when the series is truncated after a specific degree, like x? in this problem.

Double-Angle Identity

The double-angle identity is a trigonometric formula that relates the cosine of twice an angle to the square of the cosine of the original angle. This identity, specifically cos²? = ½(1 + cos 2?), is used to simplify the process of finding the power series for cosine squared by converting it into an expression involving cosines with a scaled argument.

Taylor Series Expansion

The Taylor series expansion is a method that represents a function as an infinite sum of terms calculated from the derivatives of the function at a specific point, often 0. By applying the Taylor series to the cosine function, one obtains a power series whose substitution in trigonometric identities allows for a systematic construction of the series expansion for more complex functions.

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