# Finding taylor expansion of $\cos^2x$ and $\sin^2x$

Find the taylor-series of $\cos^2x$ and $\sin^2x$.

My work so far:

We know that $\cos^2x \backslash \sin^2x = \frac{1\pm \cos 2x}{2}$, and the series for $\cos x = \sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n)!} \implies \cos 2x = \sum_{n=0}^\infty\frac{(-1)^n(2x)^{2n}}{(2n)!}$. Which leads us to $$\cos^2x \backslash\sin^2x = \frac{1\pm\sum_{n=0}^\infty\frac{(-1)^n(2x)^{2n}}{(2n)!}}{2} = \frac{1}{2}\left(1\pm\sum_{n=0}^\infty\frac{(-1)^n(2x)^{2n}}{(2n)!}\right).$$

Is it correct?

• It should at least have a $1/x^2$ term somewhere. Also note that $\frac{\cos^2x}{\sin^2x}=1-\frac1{\sin^2 x}$, which might be easier to deal with. Commented May 20, 2016 at 7:23
• @user_of_math It is not $\cot^2x$. He is running $\cos^2x,\sin^2x$ side by side. Commented May 20, 2016 at 7:24
• @Thomas. Looks ok to me, although using $/$ to indicate two alternatives is a little confusing at first glance! Commented May 20, 2016 at 7:25
• @almagest Duh, did not notice the $\pm$. Commented May 20, 2016 at 7:26
• @almagest Great, yes I'll edit / to \ :). Commented May 20, 2016 at 7:31

Since you know the Taylor expansion of $$\sin x$$ and $$\sin^2x = (\sin x)^2$$ you can use the Cauchy product: \begin{align*} \sin^2x &= \left(\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}x^{2k+1}\right)^2 \\ &= \sum_{k=0}^\infty c_k, \end{align*} where \begin{align*} c_k &= \sum_{i=0}^k \frac{(-1)^i}{(2i+1)!}\frac{(-1)^{k-i}}{(2k - 2i + 1)!} x^{2i+1}x^{2k-2i + 1} \\ & = (-1)^kx^{2k+2}\sum_{i=0}^k \frac{1}{(2i+1)!(2k+2-(2i+1))!} \\ & = \frac{(-1)^kx^{2k+2}}{(2k+2)!}\sum_{i=0}^k \frac{(2k+2)!}{(2i+1)!(2k+2-(2i+1))!} \\ & = \frac{(-1)^kx^{2k+2}}{(2k+2)!}\sum_{i=0}^k \binom{2k+2}{2i+1} \\ & = \frac{(-1)^kx^{2k+2}}{(2k+2)!} 2^{2k+1} =\frac12 \frac{(-1)^k(2x)^{2k+2}}{(2k+2)!} \end{align*} Putting it together yields \begin{align*} \sin^2x &= \frac12\sum_{k=0}^\infty \frac{(-1)^k (2x)^{2k+2}}{(2k+2)!} \end{align*} By shifting the summation index down one, we arrive at the very same thing that you derived (maybe a bit cleverer than me).
Analog $$\cos^2x$$ can be done.