To prove $$\sin{A}+\sin{B}-\cos{C}\le\frac32$$ Given $A+B+C=\pi$ and $A,B,C&gt;0$  I have managed to convert LHS to $$1-4\cos{\frac C2}\sin{\frac{A+C-B}2}\cos{\frac{A-B-C}2}$$ but that clearly isn't very helpful  One other conversion was $$\sin A+\sin B+\cos A\cos B-\sin A\sin B$$ but not sure how to proceed.  Any hints will be appreciated
If $A,B,C$ are the angle of a triangle, then show that $\sin A+\sin B-\cos C\le \dfrac32$ I tried substituting $C=180^\circ-(A+B)$ and got stuck. I also tried using the formula $\sin A+\sin B=2\sin \frac{A+B}{2}\cos \frac{A-B}{2}$ without any progress. Please help!
Why did my TeX-related GUI program stop working in Mac OS X Yosemite? HTML Element Array, name="something[]" or name="something"? I haven't logged onto my blog for years, and have, of course, forgotten the password. I click the 'Forgot' button and get a mail with a link, but that mail already says     Sorry, that key does not appear to be valid.   And sends me round the click to reset, receive a mail, click the link again etc. As if the timeout for password resets has been set to 0.5s or something.  This is a shared hosted blog, so I have full access to cPanel and all of the WP files via FTP, if that helps anything. Prove that every isometry on $\mathbb{R}^2$ is bijective Not able to calculate sum of multiple column where file data in different format .csv I have a scenario where i want to calculate sum of column   SAL|CAL|TAG|VAL 12.11"|"1.1"|"2.1"|"1.1 13.11"|"1.1"|"2.2"|"2.2 14.11"|"1.1"|"2.4"|"2.4   My below code not giving output giving output as 0.00  I need output for this column CAL VAL  awk -F'|' '{T+=$2} END { printf "%.2f\n", T }' demo.txt   output :  6.7 5.7 Infinitely number of primes in the form $4n+1$ proof Question: Are there infinitely many primes of the form $4n+3$ and $4n+1$?  My attempt: Suppose the contrary that there exist finitely many primes of the form $4n+3$, say $k+1$ of them: $3,p_1,p_2,....,p_k$  Consider $N = 4p_1p_2p_3...p_k+3$, $N$ cannot be a prime of this form. So suppose that $N$=$q_1...q_r$, where $q_i∈P$  Claim: At least one of the $q_i$'s is of the form $4n+3$:  Proof for my claim: $N$ is odd $\Rightarrow q_1,...,q_r$ are odd $\Rightarrow q_i \equiv 1\ (\text{mod }4)$ or $q_i ≡ 3\ (\text{mod }4)$  If all $q_1,...q_r$ are of the form $4n+1$, then $(4n+1)(4m+1)=16nm+4n+4m+1 = 4(\cdots) +1$  Therefore, $N=q_1...q_r = 4m+1$. But $N=4p_1..p_k+3$, i.e. $N≡3\ (\text{mod }4)$, $N$ is congruent to $1\ \text{mod }4$ which is a contradiction.  Therefore, at least one of $q_i \equiv 3\ (\text{mod }4)$. Suppose $q_j\equiv 3\ (\text{mod }4)$  $\Rightarrow$ $q_j=p_i$ for some $1\leq i \leq k$ or $q_j =3$  If $q_j=p_i≠3$ then $q_j$ | $N = 4p_1...p_k + 3 \Rightarrow q_j=3$ Contradiction!  If $q_j=3$ ($\neq p_i$, $1\leq i \leq k$) then $q_j | N = 4 p_1...p_k + 3 \Rightarrow q_j=p_t$ for some $1 \leq i \leq k$ Contradiction!  In fact, there must be also infinitely many primes of the form $4n+1$ (according to my search), but the above method does not work for its proof. I could not understand why it does not work. Could you please show me?  Regards How can I get grub2 to boot a different option only on the next boot? How do I know if I'm working on a Virtual Machine or not? Is there a way to know if the Windows machine I'm working on is virtual or physical? (I'm connecting with RDP to the machine. If it's a virtual machine it is working and handled by VMWare). Add review queues to Android app I know there are some similar posts on adding review queues to the mobile web page. However, this is regarding the Stack Exchange Android app. I really would like to review posts and edits from the Android app, however this isn't included in the app yet. I'm not sure how it would be implemented; maybe there could be a button or something for review. Could we please see this? About number truncation of ticks display in ListPlot How to show google.com in an iframe? Spambots causing server crashes. How do I block them? Tag score calculation wrong in Android App I checked yesterday and today the new homepage in the Android app. Very surprised I saw that I would almost reach a silver tag badge, that confused be because I always take a look on possible badges I can get. However I compared the score with the website and I got a total different results. See this two screenshots: Are there any concrete rules that say which words (parts of speech) in a title should start with a capital letter? What would be a correct capitalization for the title of this question? Java has a convenient split method:  String str = "The quick brown fox"; String[] results = str.split(" ");   Is there an easy way to do this in C++? gwt java.lang.ExceptionInInitializerError For $n\geq 3$ there exist $x,y\in S_n$ such that $x$ and $y$ have order $2$ and $xy$ has order $n$. Assume that $n\geq 3$. Let $S_n$ be the symmetric group with $n!$ elements. Prove that there exists $x$ and $y$ in $S_n$ such that $x$ and $y$ have order $2$ and $xy$ has order $n$.  For $n=3$, it is easy to see that $x=(1, 2)$ and $y=(1, 3)$ works. How to prove it for the general case? Let $G$ a group with normal subgroups $M,N$ such that $M\cap N=\{e\}$.   i) Show that for every $m\in M$ and for every $n\in N$, $mn=nm$.   ii) If $G$ is generated by $M\cup N$ then $G\cong M \times N$.   This is an homework question for abstract algebra. I think I can prove $i)$. But I have no idea how to prove $ii)$. Can anybody give me a hint for this one ?    Here my proof for i): Let $n \in N$ and let $m\in M$. Let $x=mnm^{-1}n^{-1}$. As $N$ is normal subgroup, $mnm^{-1} \in N$. Therefore $x \in N$.  As $M$ is normal subgroup , $nmn^{-1} \in M$. Since $m\in M$, then $x=m(nm^{-1}n^{-1})\in M$. This implies that $c\in M \cap N$, then $c\in \{e\}$, then $c=e$, and we get $mn=nm$. Create model from XYZ data points Is SO/SE about to be sold (and does that explain recent events)?