In my textbook it asks for me to: Prove that there is no constant $C$ such that $\text{arccot}(x) – \text{arctan}(\frac{1}{x}) = C $ for all $x \ne 0$. Explain why this does not violate the zero-derivative theorem. But I believe I have found such a $C$, i.e. $C =0$! I even asked WolframAlpha (http://www.wolframalpha.com/input/?i=arccot%28x%29+-+arctan%281%2Fx%29) which […]

This is from a GRE prep book, so I know the solution and process but I thought it was an interesting question: Explicitly evaluate $$\sum_{n=1}^{m}\arctan\left({\frac{1}{{n^2+n+1}}}\right).$$

$$E=\frac{1}{\sin10^\circ}-\frac{\sqrt3}{\cos10^\circ}$$ I got no idea how to find the solution to this. Can someone put me on the right track? Thank you very much.

Virtually everyone knows the basic values of the unit circle, $\sin(\pi) = 0; \ \ \sin({\pi \over 2}) = 1; \ \ \sin({\pi \over 3}) = {\sqrt{3} \over 2} \\$ And other values can be calculated through various identities, like $\sin({\pi \over 8}) =\frac{1}{2} \sqrt{2 – \sqrt{2}}\\$ Does there exist a general formula for $\sin({p\over […]

How to show the following identity holds? $$ \displaystyle\sum_{n=1}^\infty\dfrac{2z}{z^2-n^2}=\pi\cot \pi z-\dfrac{1}{z}\qquad |z|<1 $$

I know $$g(x) = \arctan(x)+\arctan(y) = \arctan\left(\frac{x+y}{1-xy}\right)$$ which follows from the formula for $\tan(x+y)$. But my question is that my book defines it to be domain specific, by which I mean, has different definitions for different domains: $$g(x) = \begin{cases}\arctan\left(\dfrac{x+y}{1-xy}\right), &xy < 1 \\[1.5ex] \pi + \arctan\left(\dfrac{x+y}{1-xy}\right), &x>0,\; y>0,\; xy>1 \\[1.5ex] -\pi + \arctan\left(\dfrac{x+y}{1-xy}\right), &x<0,\; […]

The limit is $$\lim_{x\rightarrow0} \frac{x-\sin_n(x)}{x^3},$$ where $\sin_n(x)$ is the $\sin(x)$ function composed with itself $n$ times: $$\sin_n(x) = \sin(\sin(\dots \sin(x)))$$ For $n=1$ the limit is $\frac{1}{6}$, $n=2$, the limit is $\frac{1}{3}$ and so on. Can we define a recurrent relation upon that given hypothesis? Also, how do I involve $n$ into calculation, because the final […]

$e^{i \pi} = -1$. I get why this works from a sum-of-series perspective and from an integration perspective, as in I can evaluate the integrals and find this result. However, I don’t understand it intuitively. Intuitively, what this means to me is that if you rotate pi radians around a unit circle, you will end […]

Let $S^1\subseteq\mathbb{C}$ be the unit circle and let $U\subseteq S^1$ be open. How to show that there exist a continuous function $$\theta:U\to\mathbb{R}$$ such that $$e^{i\theta(z)}=z$$ for all $z\in U$ if and only if $U\neq S^1$? I have been able to show the $\Rightarrow$ direction but not the other one. $(\Rightarrow)$ If such a continuous $\theta:S^1\to\mathbb{R}$ […]

On plotting graph for $\frac{\sin x}{x}$ using Wolfram|Alpha and Google, got that : also, I can get the value of $\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1$ using squeeze theorem and as illustrated on sources such as MIT. But I’m not able to understand how the function is defined at $x=0$ and its value came out […]

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