How would you suggest I go about solving this question? I've been thinking about it for ages and nothing comes to mind.   $$\arcsin x + \arccos x = \frac{\pi}{2}$$
The following identity is true for any given $x \in [-1,1]$: $$\arcsin(x) + \arccos(x) = \frac{\pi}{2}$$  But I don't know how to explain it. I understand that the derivative of the equation is a truth clause, but why would the following be true, intuitively?  $$\int^{x}_{C1}\frac{1\cdot dx}{\sqrt{1-x^{2}}} + \int^{x}_{C2}\frac{-1 \cdot dx}{\sqrt{1-x^{2}}} =\\  \arcsin(x) - \arcsin(C1) + \arccos(x) - \arccos(C2) = 0 \\  \text{while } \arcsin(C1) + \arccos(C2) = \frac{\pi}{2}$$  I can't find the right words to explain why this is true?    Edit #1 (25 Jan, 20:10 UTC): The following is a truth clause: $$ \begin{array}{ll} \frac{d}{dx}(\arcsin(x) + \arccos(x)) = \frac{d}{dx}\frac{\pi}{2} \\ \\ \frac{1}{\sqrt{1-x^{2}}} + \frac{-1}{\sqrt{1-x^{2}}} = 0 \end{array} $$  By integrating the last equation, using the limits $k$ (a constant) and $x$ (variable), I get the following:  $$ \begin{array}{ll} \int^x_k\frac{1}{\sqrt{1-x^{2}}}dx + \int^x_k\frac{-1}{\sqrt{1-x^{2}}}dx = \int^x_k0 \\ \\ \arcsin(x) - \arcsin(k) + \arccos(x) - \arccos(k) = m  \text{ (m is a constant)}\\ \\ \arcsin(x) + \arccos(x) = m + \arcsin(k) + \arccos(k) \\ \\ \text{Assuming that } A = m + \arcsin(k) + \arccos(k) = \frac{\pi}{2} \text{ ,for } x \in [-1,1] \end{array} $$ Using Calculus, why is that true for every $x \in [-1,1]$?    Edit #2:  A big mistake of mine was to think that $\int^x_k0 = m \text{ (m is const.)}$, but that isn't true for definite integrals.  Thus the equations from "Edit #1" should be as follows: $$ \begin{array}{ll} \int^x_k\frac{1}{\sqrt{1-x^{2}}}dx + \int^x_k\frac{-1}{\sqrt{1-x^{2}}}dx = \int^x_k0 \\ \\ \arcsin(x) - \arcsin(k) + \arccos(x) - \arccos(k) = 0\\ \\ \arcsin(x) + \arccos(x) = \arcsin(k) + \arccos(k) \\ \\ A = \arcsin(k) + \arccos(k) = \frac{\pi}{2} \text{ ,for } x \in [-1,1] \end{array} $$
I am using setInterval(fname, 10000); to call a function every 10 seconds in JavaScript. Is it possible to stop calling it on some event?   I want the user to be able to stop the repeated refresh of data. Are there any reputable studies on spanking? Let $f$ be a continuous function. If $f$ is uniformly continuous then it maps cauhy sequence into cauchy sequence. Is the converse true? That is if $f$ maps cauchy into cauchy gives the uniform continuity of $f$. I think it's no. Can suggest a counter example A question about $f(x)\equiv 0$ What is the behaviour of \fontsize if baselineskip is below a certain value? during my optimization process I found something spurious I cannot explain to myself and which I would like to understand. So with \fontsize you set the fontsize and the baselineskip. However the behaviour of the second paramater is a bit unclear to me, because if you reduce it too much, at a certain point the baselineskip gets bigger again. Let me show you my example: \documentclass[landscape]{article} \usepackage[margin=1cm]{geometry} \usepackage{lipsum,multicol}  \begin{document} \begin{multicols}{4}     \fontsize{10}{11}\selectfont     \lipsum[1]     \columnbreak          \fontsize{10}{10}\selectfont     \lipsum[1]     \columnbreak          \fontsize{10}{9}\selectfont     \lipsum[1]     \columnbreak          \fontsize{10}{8}\selectfont     \lipsum[1]     \columnbreak \end{multicols} \end{document}  If you compile this, your result will be looking like this:  Now from the first to the third column it seems logic, that the baselineskip is reduced and hence less space is needed vertically to typeset. But with the third column, even though the baselineskip is reduced even further, it needs more vertical space overall. It is strange however, that for example from line 11 to 12 (starting with tus and lum), the spacing seems to be narrower than in column 3, while in a lot of line the spacing is bigger. Changing the baselineskip to lower values down to 0 does not modify the vertical alignment further. What is the purpose of this behaviour and is it intended? I cannot see a benefit of this and I am a bit confused. Let $R$ be a commutative ring with 1 then why does $a\in N(R) \Rightarrow 1+a\in U(R)$? Let $R$ be a commutative ring with 1, we define  $$N(R):=\{ a\in R \mid  \exists k\in \mathbb{N}:a^k=0\}$$  and  $$U(R):=\{ a\in R \mid a\mbox{ is invertible} \}.$$  Could anyone help me prove that if $a\in N(R) \Rightarrow 1+a\in U(R)$?  I've been trying to construst a $b$ such that $ab=1$ rather than doing it by contradiction, as I don't see how you could go about doing that. Proving bessel identity $[J_0(x)]^2 + 2[J_1(x)]^2 + 2[J_2(x)]^2 + \cdots = 1$ and $|J_0(x)|\le 1$ The generating function for a Bessel equation is:      $$g(x,t) = e^{(x/2)(t-1/t))}$$      Using the product $g(x,t)\cdot g(x,-t)$ show that:      a) $$[J_0(x)]^2 + 2[J_1(x)]^2 + 2[J_2(x)]^2 + \cdots = 1$$      and consequently:      b)      $$|J_0(x)|\le 1, \forall x$$      c) $$|J_n(x)| \le \frac{1}{\sqrt{2}}; n=1,2,3,\cdots$$   For a) I tried the product:  $$e^{(x/2)(t-1/t))}\cdot e^{(x/2)(-t+1/t))} = 1$$  I at least arrived at the right side of the equation. Since this generates the bessel functions, I should arrive at something related to $J_n$ in the left side. I know that  $$e^{(x/2)(t-1/t))} = \sum_{n=0\infty}^{\infty} J_n(x)t^n$$  $$e^{(x/2)(-t+1/t))} = \sum_{n=0\infty}^{\infty} J_n(x)(-t)^n$$  but it's not just a matter of multiplying coefficients from the two infinite series, right?  For $b$, I tried to use  but it's not as obvious  I have no idea how to deal with $c$, could somebody help me? Computing determinant without expansion $$\begin{align}\mathrm D &amp;= \left|\begin{matrix} (b+c)^2 &amp; a^2 &amp; a^2 \\ b^2 &amp; (a+c)^2 &amp; b^2 \\ c^2 &amp; c^2 &amp; (a+b)^2 \end{matrix}\right|\\ &amp;= (a+b+c)\left|\begin{matrix} b+c - a &amp; a^2 &amp; a^2 \\ b - a -c &amp; (a+c)^2 &amp; b^2 \\ 0 &amp; c^2 &amp; (a+b)^2 \end{matrix}\right| \\ &amp;= (a+b+c)^2\left|\begin{matrix} b+c - a &amp; 0 &amp; a^2 \\ b - a -c &amp; a+c - b &amp; b^2 \\ 0 &amp; c - a-b &amp; (a+b)^2 \end{matrix}\right|\\ &amp;= (a+b+c)^2\left|\begin{matrix} b+c - a &amp; 0 &amp; a^2 \\ 0 &amp; a+c - b &amp; b^2 \\ c - a-b &amp; c - a-b &amp; (a+b)^2 \end{matrix}\right|\end{align}$$  Can $\rm D$ be further simplified without expanding ?  I feel it should be because this was competition question. When I see someone else being embarrassed / humiliated, for example a singer forgetting their words in a concert, I would say something like "I feel embarrassed for them". But is there a better word to describe this feeling?  I would say it's a mix of:   Empathetic discomfort for someone else's humiliation Tension; hope that spectators will be forgiving Frustration that I cannot help them   Other scenarios I would feel the same emotion:   Seeing someone make a presentation, the slides are all out of order, and their face gets red with embarrassment while there's a tense silence in the room. Seeing someone make an embarrassing social blunder in front of unforgiving people. Seeing a speaker who keeps making a distracting and annoying gesture, and they don't realize that the whole audience is annoyed. Seeing someone be taken advantage of, like seeing someone naively and happily accept a terrible offer on a car. Seeing a person of strong image (for example the CEO of your company) do something like accidentally leave their webcam on after a conference call, and employees see him/her let out a big burp and pick their nose or something.   In these cases it's important that the audience is not forgiving, even if they don't confront the person about it. The person may or may not know they are being humiliated.  "Pity" as proposed in an answer seems very close, but I think it lacks a few components. I think "pity" has a chronic connotation - that you are referring to the general state of things rather than a specific event.  I think "pity" also depends on the subject suffering, but in my example they are often oblivious.  It could very well be that there is no word for this at all.  NOTE: Some people do not feel the same way I do in this scenario, so you may not think this question makes any sense. Squeezing all the SEO out of a URL as possible I am working on an ecommerce site, I told our SEO consultant that I plan to make the URL scheme: /products/&lt;id&gt;/&lt;name&gt;. This is similar to Stackoverflow's URLs which are /questions/&lt;id&gt;/&lt;title&gt;.  He asked me if I could change the URL scheme to /p/&lt;id&gt;/&lt;name&gt; instead. I know why he wants this change, the word "products" isn't needed to find the correct product, and it doesn't offer any SEO, so shortening it to just p would make the relevant keywords in the &lt;name&gt; weigh more.  His main priority is maximizing SEO, but the part that I don't think he is considering is how this effects the semantics of the site. Also having the word "products" looks like it has meaning and a reason for being there, just having a p looks chaotic and ugly to me. I also don't think it makes that much of a difference does it? Stackoverflow doesn't use /q/&lt;id&gt;/&lt;title&gt; and they do just fine, I do realize that theres many factors at play here though, not just the URL.  So I want some outside opinions on which is the better way and why? How to shift EXIF date/time created by time in days, hours, minutes? Distance between surfaces I have embedded a Python interpreter in a C program.  Suppose the C program reads some bytes from a file into a char array and learns (somehow) that the bytes represent text with a certain encoding (e.g., ISO 8859-1, Windows-1252, or UTF-8).  How do I decode the contents of this char array into a Python string?  The Python string should in general be of type unicode—for instance, a 0x93 in Windows-1252 encoded input becomes a u'\u0201c'.  I have attempted to use PyString_Decode, but it always fails when there are non-ASCII characters in the string.  Here is an example that fails:  #include &lt;Python.h&gt; #include &lt;stdio.h&gt;  int main(int argc, char *argv[]) {      char c_string[] = { (char)0x93, 0 };      PyObject *py_string;       Py_Initialize();       py_string = PyString_Decode(c_string, 1, "windows_1252", "replace");      if (!py_string) {           PyErr_Print();           return 1;      }      return 0; }   The error message is UnicodeEncodeError: 'ascii' codec can't encode character u'\u201c' in position 0: ordinal not in range(128), which indicates that the ascii encoding is used even though we specify windows_1252 in the call to PyString_Decode.  The following code works around the problem by using PyString_FromString to create a Python string of the undecoded bytes, then calling its decode method:  #include &lt;Python.h&gt; #include &lt;stdio.h&gt;  int main(int argc, char *argv[]) {      char c_string[] = { (char)0x93, 0 };      PyObject *raw, *decoded;       Py_Initialize();       raw = PyString_FromString(c_string);      printf("Undecoded: ");      PyObject_Print(raw, stdout, 0);      printf("\n");      decoded = PyObject_CallMethod(raw, "decode", "s", "windows_1252");      Py_DECREF(raw);      printf("Decoded: ");      PyObject_Print(decoded, stdout, 0);      printf("\n");      return 0; } Use of the comment package How to use grep on column? I want to apply grep on particular column in column command.  myfile.txt  3,ST,ST01,3,3,856 3,ST,ST02,4,9,0234 6,N1,N101,2,3,ST 6,N1,N102,1,60,Comcast 6,N1,N103,1,2,92   My Command:  column -s, -t &lt; myfile.txt | grep -w "ST"   Here I want to only grep the pattern ST in 2nd column. How to do this ?  Expected Result:  3       ST            ST01              3            3            856 3       ST            ST02              4            9            0234 If the imaginary part of an entire function is never zero, the function is constant Let $f : \mathbb{C} \to \{z\in\mathbb{C}:\Im(z)\neq0\} $ entire . Show that $f$ is constant.  I took $g(z)=\frac{1}{f(z)}$ and I think that g is bounded, therefore it is constant (due to Louville's theorem) but how do I prove it? Is $|g(z)|=\frac{1}{|f(z)|}=\frac{1}{u^2+v^2}\le M$ valid? Why do damaged hard drives freeze the entire system? Equivalent definitions for simple connectivity Compare two dates with JavaScript Can you win a game of Splendor if you still have cards in reserve?