Find the value of $\tan(x+100) = \tan(x+50)\tan(x)\tan(x-50)$
Determine the smallest positive value of x(in degrees) for which: $\tan(x+100^{\circ}) = \tan(x+50^{\circ})\tan (x)\tan(x-50^{\circ})$
I'm a Windows/Mac/Ubuntu and Android user, so I know what I'm talking about, when I say: How do I open binary image files? (.img) They just won't open, on any OS…  I'm an Android dev… I'm currently working on a ROM, (I also program, using Windows) but I need to extract files, from .img files. I've converted them to .ext4.img but they just aren't recognized by Linux (Definitly not by Android), by Mac OS or Windows. In other words, I can't open, extract or mount them.  Can anyone help me? I'm kinda confused… Prompted by a  I was pondering the idea of bikes with variable gearing (derailleur or otherwise) but without the ability to freewheel. Have such bikes ever been widely used?  They would appear to share many of the advantages and disadvantages of fixies but be better at handling hills (up and down) while being more complex. singular/ plural in the attributive A doll house or a dolls house? Which is correct for the toy? ... There's that thing called Same Sex Couples Act; how about calling it Same Sex Couple Act?  The question is - when should I put attributives in plural? As far as I remember, Bill Bryson claimed in his Dictionary of Troublesome Words that - and that's what I vaguely recall from my classes - attributives should be in singular. Except for some exceptions. If so, what are they? I'm looking for some rule of thumb here other than "when in doubt, JFGI", of course.  NOTE: I leave out forms like a doll's house/dolls' house as they're pretty clear to me, as well as cases like The 'Big Boys and Their Motors' Club in which attributives are separate entities. How can I fix silverLight problem? I am stuck with the identity  $$ F_{2n} = \sum_{k=1}^n \binom{n}{k} F_k, $$  which happens to be formula . I am using induction, but so far without too much result.  $$ \sum_{k=1}^{n+1} \binom{n+1}{k} F_k = \\ F_{n+1} + \sum_{k=1}^n \binom{n+1}{k} F_k = \\ F_{n+1} + \sum_{k=1}^n \binom{n}{k} F_k + \sum_{k=1}^n \binom{n}{k-1}F_k =\\ F_{n+1} + F_{2n} + \sum_{k=1}^n \binom{n}{k-1}F_k, $$   I am stuck with the sum $ \sum_{k=1}^n \binom{n}{k-1}F_k$, because, on the other hand   $$ F_{n+1}+ \sum_{k=1}^n \binom{n}{k-1}F_k = F_{2n+1} $$  Which would conclude the proof. What am I missing? I want the very latest version of Python, but I don't want to bypass the apt package manager: I don't want to do an old-school source install because that can make it harder to uninstall when the official packages come out.  Python3.5 is not currently featured in any of Canonical's package repositories, which means I cannot just "apt-get install python3.5".  Is there a way to get Python3.5 installed on my PC in the meantime? it has the guide for 12.10 but not newer ones plz someone help Which are the word orders that can be found in English? Outline radius? How to set selected Dialog values to TextViews in Fragment How do I get fan control working? I know there something called fancontrol, that enables you to control the speed of your system's ventilation. I'd like to let my fans spin a bit faster as my laptop is heating up very easilly. All tutorials and stuff I've found are for old versions of Ubuntu and don't seem to be working anymore.  Can anyone explain to me or give me a good link on how I  can get it working on Ubuntu? Something different with the same effect is also fine. Is it possible to enforce unique values in a column, excluding nulls? Should there always be a comma after "therefore","However" etc.? Trouble downloading packages list due to a "Hash sum mismatch" error How do you do load testing and capacity planning for databases? This is a  about capacity planning for databases. Related:      I'm looking to create a canonical question of tools and methods of capacity planning for databases. This is intended to be a canonical question. Obviously, the general workflow is:  Put your scenario in place Add monitoring Add traffic Evaluate results Remediate based on results Rinse, repeat until reasonably happy  Please feel free to describe different tools and techniques for different web-servers, frameworks, etc., as well as best-practices. Dynamic date range query I'm trying to create two cells that will update my query: A >= date"A1" and A &lt;= date "B1".  My goal is to create a search page for the data, where the end user could insert a date or the =today() function and it would update the query. I am currently getting a formula parse error from:  =QUERY(Data!6:1397,"SELECT A,B,C,D,E,F,G,H,I,J,K,L WHERE A &gt;= date"A1"and  A &lt;= date"B1"") Can't see a glass object through another glass object in Eevee I was just testing Eevee and found that you cannot view a glass object through another glass. Is there any workaround? It works fine only If I rotate the glass window or put my camera to the back side of the scene (Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)  For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x &gt; 0$.  Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining  $$x_{n+1} = \sqrt{c+x_n}$$  Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation.  Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$.    This is being repurposed in an effort to cut down on duplicates, see here: .  and here: . SQL group by day, show orders for each day I have an SQL 2005 table, let's call it Orders, in the format:  OrderID, OrderDate,  OrderAmount 1,       25/11/2008, 10 2,       25/11/2008, 2 3,       30/1002008, 5   Then I need to produce a report table showing the ordered amount on each day in the last 7 days:  Day,        OrderCount, OrderAmount 25/11/2008, 2,          12 26/11/2008, 0,          0  27/11/2008, 0,          0  28/11/2008, 0,          0 29/11/2008, 0,          0 30/11/2008, 1,          5   The SQL query that would normally produce this:  select count(*), sum(OrderAmount)     from Orders     where OrderDate&gt;getdate()-7     group by datepart(day,OrderDate)   Has a problem in that it will skip the days where there are no orders:  Day,        OrderCount, OrderAmount 25/11/2008, 2,          12 30/11/2008, 1,          5   Normally I would fix this using a tally table and outer join against rows there, but I'm really looking for a simpler or more efficient solution for this. It seems like such a common requirement for a report query that some elegant solution should be available for this already.   So: 1. Can this result be obtain from a simple query without using tally tables?  and 2. If no, can we create this tally table (reliably) on the fly (I can create a tally table using CTE but recursion stack limits me to 100 rows)? OK...before everyone blasts this with references to the relativistic invariance of the physical laws, time dilation, etc let me add some context. Also, I am an amateur with an interest in physics, so I don't know the details of the physics. However, I have read enough about the cosmological principle and cosmic time to develop this "confusion" .  Based on evidence gathered to-date, it appears that the universe is both , as evidenced by observations and the . This principle, when combined with the FLRW metric, allows space-time to be divided into non-intersecting slices and therefore establish a "cosmic time."   Now, the existance of such a time in no way suggests that it is the "right" time or a true "now", hence violating relativity -- although some presentist philosophers (e.g., W.L. Craig) have tried to make this argument, but it is not generally accepted in philosophical circles.  However, this leaves me confused about an apparent disconnect between relativity and the cosmological principle:  Isotropy only holds if we are at rest relative to the cosmic microwave background radiation (universal rest frame), o/w anisotropy is present - yet each reference frame is supposed to allow equally valid observations. How can these two be reconciled when most reference frames would lead us to conclude that the universe is NOT isotropic?  It seems that only by appealing to the idea of being at rest relative to "universal rest frame" can we explain away any discrepancies from isotropy as due to "peculiar motion". However, doesn't this give this "universal rest frame" and its associated time an empirically privileged status, even though physical laws work just fine in every reference frame?   For example (pardon any abuse of astronomy): if look off in some part of space and see only quasars, and then in another part of space and see only brown dwarfs, but I measure both as apparently the same distance from me, then can I conclude that we are in motion, since otherwise we would have an empirical contradiction (kind of like finding dinosaur and human remains in the same strata)?  Any help on where I am going wrong would be helpful. Intuitively, I don't think there should be a way to establish absolute ordering between space-like separated events, but the above reasoning suggests otherwise.