Someone told me there is an equation $$\int_0^{\pi/2}f(sinx)dx=\int_0^{\pi/2}f(cosx)dx$$ With this equation, it's easy to get the answer$\frac{\pi}{4}$. What I want to know is why we have this equation, and if there is an alternative way to integrate it?
Calculating with Mathematica, one can have $$\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}\,\mathrm dt=\frac{\pi}{4}.$$   How can I get this formula by hand? Is there any simpler idea than using $u = \sin t$?     Is there a simple way to calculate $$ \int_0^{\pi/2}\frac{\sin^n t}{\sin^n t+\cos^n t}\,\mathrm dt $$ for $n&gt;3$? Could anyone come up with a reference for this exercise?
My (very simple) question to a friend was how do I prove the following using basic trig principles:  $\tan75^\circ = 2 + \sqrt{3}$  He gave this proof (via a text message!)  $1. \tan75^\circ$   $2. = \tan(60^\circ + (30/2)^\circ)$  $3. = (\tan60^\circ + \tan(30/2)^\circ) / (1 - \tan60^\circ \tan(30/2)^\circ) $  $4. \tan (30/2)^\circ = \dfrac{(1 - \cos30^\circ)}{ \sin30^\circ}$    Can this be explained more succinctly as I'm new to trigonometry and a little lost after (2.) ?    EDIT  Using the answers given I'm almost there:   $\tan75^\circ$    $\tan(45^\circ + 30^\circ)$   $\sin(45^\circ + 30^\circ) / \cos(45^\circ + 30^\circ)$   $(\sin30^\circ.\cos45^\circ + \sin45^\circ.\cos30^\circ) / (\cos30^\circ.\cos45^\circ - \sin45^\circ.\sin30^\circ)$    $\dfrac{(1/2\sqrt{2}) + (3/2\sqrt{2})}{(3/2\sqrt{2}) - (1/2\sqrt{2})}$   $\dfrac{(1 + \sqrt{3})}{(\sqrt{3}) - 1}$   multiply throughout by $(\sqrt{3}) + 1)$   Another alternative approach:   $\tan75^\circ$    $\tan(45^\circ + 30^\circ)$   $\dfrac{\tan45^\circ + \tan30^\circ}{1-\tan45^\circ.\tan30^\circ}$   $\dfrac{1 + 1/\sqrt{3}}{1-1/\sqrt{3}}$   at point 6 in above alternative Can my friend and I play minecraft together with different mods? Simplify a combinatorial sum Why is $\sum_{k=0}^n (_k^n) 2^k$ simplified as $3^n$ and not just $2^n$ ? The answer uses the binomial theorem as a solution: let $a=1$ and let $b=2$. So $1+2 = 3$. But since $1+2 = 3$ why do we not use $1+1 = 2$? Need to give permissions to /tmp/cache and /tmp/templates_c but can't find them According to preliminary estimates, the project should be completed by 2020 at a cost of about RUB 40 bn.    or      According to preliminary estimates, the project should be completed by 2020 at the cost of about RUB 40 bn.    Are both options okay, or is the first better?   We are introducing the sum to the reader, so logically it should be the first option. But there's a lot of google-hits for "at the cost of" too. Page number in title page I want to add page number from 531 to 536 in my latex IEEE conference template. But page number appearing from page 2 i.e. 532.  \documentclass[conference]{IEEEtran} %\IEEEoverridecommandlockouts  \makeatletter \let\old@ps@headings\ps@headings \let\old@ps@IEEEtitlepagestyle\ps@IEEEtitlepagestyle \def\confheader#1{%     % for all pages except the first     \def\ps@headings{%         \old@ps@headings%         \def\@oddhead{\strut\hfill#1\hfill\strut}%         \def\@evenhead{\strut\hfill#1\hfill\strut}%     }%     % for the first page     \def\ps@IEEEtitlepagestyle{%         \old@ps@IEEEtitlepagestyle%         \def\@oddhead{\strut\hfill#1\hfill\strut}%         \def\@evenhead{\strut\hfill#1\hfill\strut}%     }%     \ps@headings% } \makeatother  \confheader{ 2020 IEEE International Conference on Computation, Automation and Knowledge Management (ICCAKM) }   %%footer %\usepackage[pscoord]{eso-pic} %\newcommand{\placetextbox}[3]{ %   \setbox0=\hbox{#3} %   \AddToShipoutPictureFG*{ \put(\LenToUnit{#1\paperwidth},\LenToUnit{#2\paperheight}){\vtop{{\null}\makebox[0pt][c]{#3}}} %   } %} %\placetextbox{.23}{0.055}{\small{978-1-5386-8007-0/19/\$31.00~\copyright 2019 IEEE}}     \usepackage{cite} \usepackage{amsmath,amssymb,amsfonts} \usepackage{algorithmic} \usepackage{graphicx} \usepackage{textcomp} \usepackage{xcolor} \usepackage{float} \usepackage[export]{adjustbox} \usepackage{subcaption}     % %\usepackage[pscoord]{eso-pic} %\newcommand{\placetextbox}[3]{ %   \setbox0=\hbox{#3} %   \AddToShipoutPictureFG{ \put(\LenToUnit{#1\paperwidth},\LenToUnit{#2\paperheight}){\vtop{{\null}\makebox[0pt][c]{#3}}} %   } %} %\placetextbox{.2}{0.055}{978-1-4799-5522-0/15/\$31.00~\copyright 2015 IEEE}   \IEEEoverridecommandlockouts \IEEEpubid{\makebox[\columnwidth]{978-1-7281-0666-3/20/\$31.00~\copyright~2020         IEEE \hfill} \hspace{\columnsep}\makebox[\columnwidth]{ }}      \thispagestyle{plain}     \pagestyle{plain}     \setcounter{page}{531}   \begin{document}  \title{Performance Analysis of Physically     Unclonable Function Based on Voltage Divider Arrays} \author{\IEEEauthorblockN{Sumit Chaudhary$^{1}$}     \IEEEauthorblockA{\textit{Dept. of Electronics and Communication} \\         \textit{MNIT Jaipur, India}\\         2016pev5165@mnit.ac.in}     \and     \IEEEauthorblockN{Chitrakant Sahu$^{2}$}     \IEEEauthorblockA{\textit{Dept. of Electronics and Communication} \\         \textit{MNIT Jaipur, India}\\         chitrakant.ece@mnit.ac.in}}  \maketitle   \end{document} So, according to Einstein's theory of general relativity, gravity is not a force instead it is a consequence of objects with mass deforming spacetime, right? And so, according to him, there is no difference between standing on a platform accelerating upwards at 9.81 m/s^2 and standing on Earth, right? But, there is a difference. At least, I think it's a valid difference. If you drop two apples from a high altitude on Earth, not only will they accelerate downwards at 9.81m/s^2 but they will also slightly move towards each other because the Earth is a sphere and gravity acts radially inwards pulling the two apples closer to each other. But if you drop the apples from the same height above the accelerating platform, the apples will only appear to accelerate downwards at 9.81 m/s^2 but will not move towards each other like they would on Earth. Why is this true? Is this even a valid difference? If it isn't, why isn't it? Is there a string math evaluator in .NET? If I have a string with a valid math expression such as:  String s = "1 + 2 * 7";   Is there a built in library/function in .NET that will parse and evaluate that expression for me and return the result? In this case 15. We're using list view and details OnClick Javascript buttons which reference custom js ans css files stored in static resources and containing some shared logic/styles. The sample code below demonstrates how they are referenced.  {!REQUIRESCRIPT('/resource/CustomJQuery/js/jquery-1.10.2.js')}  {!REQUIRESCRIPT('/resource/CustomLib')}  $("head").append($("&lt;link rel='stylesheet' href='/resource/CustomJQuery/css/smoothness/jquery-ui-1.10.4.custom.css' type='text/css' media='screen' /&gt;"));   These files are also referenced in Visualforce pages.  Introduction of "Serve Static Resources from force.com domain" critical update breaking this way of using static resources in OnClick Javascript buttons - there is also Known Issue created for this .  So, my questions are:  1) Are there any ways to make it work with "Serve Static Resources from force.com domain" critical update activated. My assumptions for now are below, but they seem to be risky and may require significant changes:   hosting these js and css files on own server and referencing them in OnClick Javascript buttons and all other places; move to custom visualforce pages;   2) Does anyone know what's going on with "Serve Static Resources from force.com domain" critical update right now? It was scheduled in Winter 15 first, then it was postponed to Summer 15. Now it can't be found in any org in list of all critical updates at all. It has even disappeared from the list in orgs where it used to be 1-2 weeks ago.  3) How long Known Issue can be "In review" status, since related one has this status for more than 2 months?  Thank you How do you select elements based on their style? Using jQuery, how would you find elements which have a particular style (eg: float: left), regardless of whether it's an inline style or one defined in a CSS file? Inequality $\left(a-1+\frac{1}b\right)\left(b-1+\frac{1}c\right)\left(c-1+\frac{1}a\right)\leq1$ USB sticks and multiple partitions I've been wondering about this for a while, since getting interested in knife sharpening.  When using a knife, it will get duller. One (the primary) reason being that you "bend" the edge. You fix this using a honing steel.  After a while though, you need to sharpen the knife again.  Another setting: If you have a brittle knife and/or mistreat your knife you will chip it.   As far as I can tell, this should mean that small pieces of metal (dust, shavings, etc.) come lose, and then likely get in the food.  Is this correct? Are there no health considerations when digesting small pieces of very sharp metal? Finding an expression for $\tan (a+ b + y)$ which involves only $\tan a, \tan b,$ and $\tan y$. Let $ \mathbb R _ { &gt; 0 } $ be the set of positive real numbers. Find all functions $ f : \mathbb R _ { &gt; 0 } \to \mathbb R _ { &gt; 0 } $ such that $$ f \big ( x y + f ( x ) \big) = f \big( f ( x ) f ( y ) \big) + x $$ for all positive real numbers $ x $ and $ y $.  What I thought: We could change $ x $ by $ y $, and then subtract. Source: Brazil National Olympiad 2019 #3 Arc length parameterization lying on a sphere Center figure that is wider than \textwidth I have a figure that is wider than the \textwidth of my document. (I don't want to change the \textwidth.) As a result, the leftmost part of the figure is flush with the left margin, while the rightmost part spills into the right margin too much. I'd like to center the figure, so that it will exceed the \textwidth equally on both sides. Find $cot\space x_n$ if $sin(x_{n+1}-x_n)+2^{-(n+1)}sin\space x_nsin \space x_{n+1}=0$ for $n\ge1$ Also Given     Suppose $x_1=tan^{-1}2&gt;x_2&gt;x_3&gt;......$ are positive real numbers   I cannot understand as to how I should approach but I have a hunch which is a bit hand-wavy way to prove it but tried to show that  $lim_{n \to \infty}x_n =\dfrac{\pi}{4}$ (For the proof see my edits in @hypernova's answer)  But can anybody prove it and then use it to find $cot \space x_n$ please? How do I run a game for a larger group? I am currently running a Dungeons and Dragons 4 campaign for a group of 9 players.  Whittling the group down to a more manageable size is not really an option since they are all family.  When we do combat it takes forever even if I have one player help me and allow them to go 2 at a time.  When we do story based adventure there are 2-3 players that do most of the talking and the others twiddle their thumbs.  I've tried encounters with many foes, encounters with only a few tough foes, dialog only encounters, social puzzle encounters, everything I can think of.  What else can I try to get this group more engaged? I just discovered an incredibly annoying "feature" of Windows 7 tonight.  If you click on your optical drive in Explorer, and there is no disc in the drive, Windows 7 is kind enough to open the tray for you.  I'm finding it hard to contain my rage over this idiotic design decision.  WTF?  According to , this behavior was apparently introduced in Vista.  The author posted a workaround, but I don't want a workaround.  I want it to stop doing this.  Like he, I have a case with a door that covers the drive bays.  If it's closed, and I accidentally click on that optical drive, it knocks on that door 10 or 12 times before giving up.  Can this behavior be turned off in Windows 7?