I am not able to show that:     If $x+y+z=\pi$, show that $\tan(x) + \tan(y) + \tan(z) = \tan(x) \tan(y) \tan(z)$.
I want to prove  \begin{equation*} \tan A  + \tan B + \tan C = \tan A\tan B\tan C \quad\text{when } A+B+C = 180^\circ \end{equation*}  We know that   \begin{equation*} \tan(A+B) = \frac{\tan A+\tan B}{1-\tan A\tan B}~\text{and that}~A+B = 180^\circ-C. \end{equation*}  Therefore $\tan(A+B) = -\tan C.$ From here, I get stuck. Please help.
Why is logistic regression a linear classifier? Since we are using the logistic function to transform a linear combination of the input into a non-linear output, how can logistic regression be considered a linear classifier?  Linear regression is just like a neural network without the hidden layer, so why are neural networks considered non-linear classifiers and logistic regression is linear? My textbook states that:  ...The capacitance $C$ depends only on the geometrical configuration (shape, size, separation)of the system of two conductors. [As we shall see, later, it also depends on the nature of the insulator (dielectric) separating the two conductors.]...  Now I wanted to know if these were the only two factors on which the capacitance of a conductor depends. So I tried thinking about it from the basic definition, which is:  Capacitance is the ratio of the change in electric charge of a system to the corresponding change in its electric potential. (Source: )  i.e., $$C= \frac {dQ}{dV}$$ where, $V$ is the potential of the conductor with respect to zero potential at infinity. Now, I can not think of why $C = f(V)$ isn't a possible scenario, assuming that shape size, etc remains constant. So:  Why cannot $C$ be a function of $V$? Find the value of $\int_{-\infty}^{\infty} \frac{e^{-x^2}}{1+x^2} dx$ . Find the value of $\int_{-\infty}^{\infty} \frac{e^{-x^2}}{1+x^2} dx$     My attempt  First, I know the existence of this integral since $$ \mid \int_{-a}^{a} \frac{e^{-x^2}}{1+x^2} dx \mid \;\; \leq\;\;  \mid \int_{-a}^{a} \frac{1}{1+x^2} dx \mid$$  Next, let $R &gt;0 $ be an arbitrary and $\gamma_1(t) = -R(1-t) + Rt , \;\; 0 \leq t \leq1$ and $\gamma_2(t)= Re^{i\pi (t-1)}, \;\; 1 \leq t \leq 2 $.  So, take $f(z) = \frac{e^{-z^2}}{1+z^2}$.   I tried to find the value or upper bound of norm of $$\int_{\gamma_2} \frac{e^{-z^2}}{1+z^2} dz$$.  However, $$ \lvert\int_{\gamma_2} \frac{e^{-z^2}}{1+z^2} dz\rvert \leq  \frac{\text{max}\lvert e^{-z^2}\rvert}{R^2-1}\pi R = \frac{e^{R^2}}{R^2-1}\pi R $$ So, I can't proceed this.  May I ask you how to solve this? Java Generics: List, List, List Number of solutions of an arithmetic function's equation Say, an equation is given below \begin{equation} 2\pi(x) - \pi(2x)=\omega(x) \end{equation}  where $x$ is a positive integer, $\pi(x)$ is the prime-counting function, and $\omega(x)$ is the number of distinct prime factors of $x$.  I would like to know if this equation holds for finitely many values of $x$ or infinitely many.  If one uses asymptotic formula (from number theory) it can be written that \begin{equation} \omega(x) \sim \ln x / \ln \ln(x) \end{equation} (from )  and  \begin{equation} \pi(x) \sim x/\ln(x), \end{equation} so the above equation can be rewritten (after simplification) as \begin{equation} 2x\ln(2)=\ln(x)\ln\ln(x)\ln(2x) +f(x) \end{equation}  where $f(x)$ is an error term.  If  I plot left side (which is linear, supposed to be a straight line) and right side(a curve),assuming $ f(x) $ is 100% accurate, then the equation implies(?!) that the right will intersect left side. Since line can intersect a curve ($ln  $ function) finite times, so there are finite intersection point. So, can I say that there are only finitely many values of $x$?  graph example-   here f(x) is assumed as x + sinx  without the f(x)- Given $f(x) = x + |x|$ for what values of $x$ is $f$ differentiable Problem : Given $f(x) = x + |x|$ for what values of $x$ is $f$ differentiable?   For the sake of generality, let's assume that it is unknown to us that $|x|$ is not differentiable at $x = 0$  Attempted Solution :  Using the definition of differentiability, a function is differentiable over an interval $I$ $\text{iff}$  $$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}, \ \ \forall x\in I$$  Now implicitly this definition of differentiability requires $$\lim_{h\to 0^+} \frac{f(x+h)-f(x)}{h} = \lim_{h\to 0^-} \frac{f(x+h)-f(x)}{h}$$  Therefore $f$ will be differentiable only when  $$\lim_{h\to 0^+} \frac{(x + h + |x+h|)-(x + |x|)}{h} = \lim_{h\to 0^-} \frac{(x - h + |x-h|)-(x + |x|)}{h}$$  But it is unclear what to do next as $x$ is just arbitrary $$ \begin{equation} \begin{aligned} |x+h| &amp;= \begin{cases}x+h &amp; \text{if} &amp; x \geq -h\\ -x-h &amp; \text{if} &amp; x &lt; -h \end{cases} \\ &amp;= \begin{cases}x+h &amp; \text{if} &amp; x \geq 0\\ -x-h &amp; \text{if} &amp; x &lt; 0 \end{cases} \end{aligned} \end{equation}$$    A Wrong Solution :  We could take the derivative of $f$, (and evaluate the domain of the derivative $f'(x)$ to values for which it is defined), using the rules of differentiation, but we would get a wrong answer.  $$f'(x) = \frac{|x| + x}{|x|}$$ $$\implies f'(x) = \begin{cases} 2 &amp; \text{if} &amp; x \geq 0 \\ 0 &amp; \text{if} &amp; x &lt; 0 \end{cases}$$  This implies the derivative $f'(0)$ exists, when it does not as per the definition of differentiability. Why is that so?    Questions:   How can a solution be found using the definition of differentiablity? Why does taking the derivative of $f$, $f'(x)$ and evaluating it's domain, not give the correct values of $x$ for which $f$ is differentiable? How to build Blender as a Python module? I have seen this question now many times and every question here and on other forum points to  experimental tutorial by . Now sadly a lot of answers are invalid right now because this central link is down. I couldn't find anything by Googling so if anybody has a fork of the tutorial or a second source that would be great? The link can't be gone for too long, I saw an answer with it 4 months ago.  The only other thing I have found is , but it also references the now missing tutorial. Any ideas how to find it? Can I stay overnight [15-hour layover] in Haneda airport without a Japan visa? I’m a Filipino citizen. My flight goes from Vancouver to Manila via Tokyo Haneda. I will arrive in Japan at 6pm then depart at 9am the next day. I'm not planning to go out of the airport during my layover, so I don't see a need to have a transit visa. However, I called the Consulate-General of Japan here in Canada, and they asked me to get a transit visa.  Then I called the Ministry of Foreign Affairs in Japan and asked them the same question.. at first, they told me I don’t need a visa as long as I stay inside the airport, but when I told them that the Consulate-General of Japan in Canada asked me to get a visa, they also asked me to get one.  I'm so confused. I would like to know your knowledge or experience regarding this. How do I find out which process is locking a file using .NET? What is the difference between logistic regression and Fractional response regression? strtok segmentation fault Counting the number of surjections. How many functions from set $\{1,2,3,\ldots,n\}$ to $\{A,B,C\}$ are surjections? $n \geq 3$  Attempt  I was hoping to count the number of surjections by treating $A,B,C$ like bins, and counting the number of ways to fill them with $1,2,3,\ldots,n$ such that no bins are empty. This is equivalent to putting two "separator" bars in between the $n$ numbers or $n-1 \choose 2$.  However, I think I am missing a step here: prior to putting in the separator bars, I should permutate the $n$ elements. So, multiply by $n!$.  The order of elements in the same bins do not matter. So I should be divided by the number of permutations of the elements in each bins. This is where I am stuck.  Additional Info  I made the question up myself. Feel free to modify the question into a more solvable form if it is too difficult/complicated. Which derivation of drift velocity is correct? In the derivation of drift velocity I have seen two variations and want to know which one's correct.   $s=ut+\frac{at^2}{2}$ Assume that the drift velocity of any electron in any conductor is : $$\vec{v_d}=\frac{\vec{l}}{t}$$ Due to the electric field the acceleration of electrons in any conductor is: $$a=\frac{-e\vec{E}}{m}$$ Now the distance travelled by an electron after a long time (initial thermal velocity = 0) $$l=\frac{at^2}{2}\implies \frac{-e\vec{E}t^2}{2m}$$ the time between the collisions is $\tau$ $\therefore$ $$l=\frac{-e\vec{E}\tau^2}{2m}$$ thus the velocity is $$\vec{v_d}=\frac{-e\vec{E}\tau}{2m}$$ In another proof I saw the author using $v=u+at \implies v_d=\frac{-e\vec{E}\tau}{m}$   My question is which of the the two equations of motion can be used in the proof? Can they be used at all. Is there a way to determine the respective canonical status of Tolkien's Middle-earth related works? It is almost always difficult to determine the canonicity (if that's a word) of different works within a fictional universe, but the problems are even more complicated in regards to Tolkien's Middle-earth/Arda related material.  It is safe to say that LotR, the second edition of The Hobbit, and probably the published Silmarillion are canon.  But even the published Silmarillion is, in large part, the work of Christopher Tolkien, although to a lesser degree than it is the work of his more famous father.    When we look at the rest of the corpus, things become even more opaque.  Unfinished Tales contains different versions of the same stories.  The same is true, in even greater degree, of The History of Middle-earth.  These volumes might be considered an almost equal endeavor on the parts of Christopher and J.R.R Tolkien.    Is there a tentative canon of Tolkien's Middle-earth related works, and if not, is it possible to create one?    With such a large body of work, of varying degrees of canonicity, it would be very helpful to have something like the now-outdated levels of canon associated with the Star Wars universe, ranked by a series of letters.  Has anyone put together such a system for Tolkien's Middle-earth related works? Regex for splitting a string using space when not surrounded by single or double quotes How can a capacitive touch screen be triggered without human contact? I want to robotically touch an iPhone screen without any human intervention.   I've done some experiments but have not found a reliable solution. It seems even styluses that work through gloves still rely on the capacitive characteristics of the human body.    Carrot held by human works Carrot held in plastic clamp doesn't Pogo Stylus works held by human and held by human through clothes Stylus does not work held in plastic clamp.  Stylus works held in plastic clamp attached to human through clothes via jump lead. Stylus held in plastic clamp with jump lead dangling off seems to work most the time.   Is it possible to mimic the human element using capacitors and other components? What signal does the capacitive screen need to be triggered?  Thanks Stuck at 0% [waiting for headers] On googling this error it seems that a lot of people are having this issue.  apt-get upgrade hangs at 0% [waiting for headers] apt-get install apache2 hangs at 0% [waiting for headers]. Some apt-get commands are working, but only certain packages.  Any ideas? I don't think this is an issue with slow repos because I've had it up for a long while and it hasn't budged from 0%. I have edited sources.list but I removed the lines I added when this issue presented itself and it hasn't made a difference.  Any clues would be really useful. This is one thing that keeps bugging me, and maybe there's a direct answer. Grammatically, which one is more correct of these two?  Does it make a difference?  I tried not to do that. I tried to not do that. The problem Comments are terrible; no one agrees on what they're for, or how long they should be kept. Comment flags are equally terrible, because if no one's on the same page as far as what should be posted in the first place then no one can agree on what should be removed either. On top of that, we have this confusing set of flags where two mean &quot;noise&quot;, one means &quot;vile&quot;, one means &quot;moderator, help!&quot; and one means nothing at all - the end result is that it's all but impossible to even analyze comment flagging. The solution (or, part of it) Two years ago, in a fit of rage, I . Nothing got done. As always, everyone has a different idea of how things are supposed to work, and I'd bitten off more than I could chew. So, here we are, two years later, and comment flags still suck. So let's start by just cutting the fat: if we can trim down the list of flags to a set of clearly distinct problems, we might have something we can build on. Here's what I'm proposing, based on the past two years of discussion:   Re-word the &quot;rude or offensive&quot; flag to read,  rude or abusive: this comment violates .   Remove the &quot;not constructive&quot;, &quot;obsolete&quot; and &quot;too chatty&quot; flags.  Add a new comment flag:  no longer needed: this comment is obsolete, chatty or otherwise unnecessary   Re-word the &quot;other...&quot; flag:  in need of moderator attention: these comments require action by a moderator. Be specific and detailed!    That's it; that's all I'm proposing to start out - simplify things so that we have something to build on. See also: Should we restrict signup for a service to Facebook accounts only?