User:Thatha

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This page is under heavy construction! For the time being visit my website.

Languages, Scripts, and Grammar

G	This user is a student or alumnus of Grinnell College.

en	This user is a native speaker of English.

el	Αυτός ο χρήστης μιλά ως μητρική γλώσσα την ελληνική.

grc-3

Ὅδε ὁ χρήστης δύναται συνεισφέρειν ἀνωτέρᾳ γνώσει τῆς ἀρχαίας ἑλληνικῆς.

de-2

Dieser Benutzer hat fortgeschrittene Deutschkenntnisse.

fr-1

Cet utilisateur peut contribuer avec un niveau élémentaire de français.

ja-1

この利用者は少しだけ日本語を話すことができます。

ipa-3
ə

This user has an advanced understanding of the International Phonetic Alphabet.

~~mankind~~

This user supports the use of gender-neutral language.

they

This user considers singular they standard English usage.

snkt

This user says sneaked.

infin¦
itive

This user chooses never to split infinitives.

by	The passive voice may be used by this user.

…in.

Ending a sentence with a preposition is something that this user is okay with.

whom

This user insists upon using whom wherever it is called for, and fixes the errors of whomever he sees.

which & that

This user knows how to use which and that correctly.

“,;:’

This user is a punctuation stickler.

. The

This user does not put two spaces after a full stop.

,	This user fixes comma splices. They are annoying.

A, B, and C

This user prefers the serial comma.

Mix

This user has been influenced by too many dialects of English to use one orthography, vocabulary and grammar consistently.

About Me

This user is 19 years old.

This user comes from Greece.

Computing

AIM-Able

This user understands AIM talk perfectly well, but does not seriously use it. Ever.

C-2

This user is an intermediate C programmer.

asp

This user can code in ASP.

asm

This user eats, breathes and sleeps in assembler.

C++

This user can program in C++.

Java

This user can program in Java.

pas

This user can program in Pascal.

php

This user can program in PHP.

This user can program in Scheme.

This user is a Rubyist.

bash

This user is a bash scripter.

<html>

This user can write HTML.

js	This user can program in JavaScript.

This user can typeset using LaTeX.

mysql

This user writes programs that access MySQL.

re	This user is a regular expression programmer.

xhtml

This user can write XHTML.

prog-4

This user is an expert programmer.

@	This user can be reached by email.

This user's AOL Instant Messenger screenname is thatha7777.

This user uses Jabber as thatha@jabber.org.

	This user uses Skype. His/Her username is thatha7777.

Y!	This user's Yahoo! Messenger screename is thatha7777.

This user has a website, which can be found here.

dA	This user is a deviant as

Digg	This user contributes to Digg.

f	This user maintains a Facebook profile.

last
·fm

This user tracks what they listen to on Last.fm.

This user maintains a blog.

This user uses WordPress.

This user uses Google as a primary search engine.

OWN

This user uses an email account with his/her own domain name.

This user uses Adium to chat on various instant messaging networks.

This user contributes using The GIMP.

This user prefers Mozilla Firefox.

This user contributes using Safari.

This user runs Mac OS X.

g	This user contributes using Gentoo Linux.

This user contributes with Red Hat Linux.

This user contributes using Debian GNU/Linux.

This user contributes with GNOME.

This user contributes using Microsoft Windows .

This user contributes using Nano.

This user contributes using Vim.

This user writes with the OpenOffice.org suite.

This user contributes using a PDA.

This user contributes using a laptop.

intel inside This user contributes with a computer based on the Intel Core Duo microprocessor. CORE DUO

This user contributes using an Apple Mac mini computer.

This user contributes using a .

BB	This user contributes using a broadband connection.

Preferences

This user is a carnivore.

This user believes that Caffeine is necessary in large doses daily.

This user drinks hot chocolate.

This user drinks orange juice.

Template:User H20

This user drinks tea.

This user believes that you should never put milk in with the tea bag.

This user likes the sweet flavour of the Piña Coladas.

CHOH-2

This user drinks occasionally.

cvg-0

This user does not play or is not interested in video games.

This user plays games on a graphing calculator.

This user is interested in ancient civilizations.

This user is interested in the ancient Celts.

This user is interested in Ancient Greece.

$\sum_{n=0}^{\infty} \frac{4{(-1)^n}}{2n+1} = \pi$

This user is an advanced mathematician.

λ	This user is fluent in Lambda calculus.

This user uses the metric system and loves it.

Pi ≈ 3.14159265358979323846264

33832795028841971693993751058 20974944592307816406286208998 62803482534211706798214808651 32823066470938446095505822317 25359408128481117450284102701 93852110555964462294895493038 19644288109756659334461284756 48233786783165271201909145648 56692346038610454326648213393 60726024914127372458700660631 55817488152092096282925409171 53643678925903600113305305488 20466521384146951941511609

This user plays the flute.

This user plays the piano.

This user plays the recorder.

This user listens to World Music

This user enjoys classical music.

This user enjoys pop music.

This user enjoys rock music.

This user enjoys
heavy metal music.

This user enjoys indie rock.

This user is an audiophile.

iTS	This user uses the iTunes Store.

Cass

This user knows that you gotta make your own kind of music.

This user is interested in Linguistics.

This user misses Pluto. May its planethood rest in peace.

This user enjoys electronics or is an electronics hobbyist.

'	This user has the knack.

This user prefers .

This user enjoys pistol shooting.

This user is a bicyclist.

This user couldn't care less about .

This user is a bibliophile.

This user is interested in Criminology

This user does not like socklessness.

This user is interested in law.

This user is an art lover.

This user enjoys cooking.

This user enjoys photography.

This user enjoys writing.

This user is a lover.

This user's home is overrun with cats.

This user is a member of the worldwide family of Scouts.

Calculating $\int \csc{x} \;dx$

$\int \csc x \;dx$ = $\int \frac{1}{sin{x}} \;dx$ = $\int \frac{1}{2 \sin{\frac{x}{2}} \cos{\frac{x}{2}}} \;dx$ = $\frac{1}{2} \int \frac{1}{ \sin{\frac{x}{2}} \cos{\frac{x}{2}}} \;dx$ , since $\sin{\theta}=2 sin{\frac{\theta}{2}} cos{\frac{\theta}{2}}$ (Double-angle formulae).

$\frac{d}{dx} \ln{|\tan{x}|}$ = $\frac{1+\tan^2{x}}{\tan{x}}$ = $\frac{\sec^2{x}}{\tan{x}}$ = $\frac{1}{\cos^2{x}\tan{x}}$ = $\frac{1}{\cos^2{x}\frac{\sin{x}}{\cos{x}}}$ = $\frac{1}{\sin{x}\cos{x}}$ , thus $\int \frac{1}{ \sin{x} \cos{x}} + C \;dx$ = $ln | tan x |$ , where $C$ is a constant of integration.

Thus, $\frac{1}{2} \int \frac{1}{ \sin{\frac{x}{2}} \cos{\frac{x}{2}}} \;dx$ = $\frac{1}{2} 2 \ln{|\tan{\frac{x}{2}}|} + C$ = $\ln{|\tan{\frac{x}{2}}|} + C$ .

To summarize, $\int \csc{x} \;dx$ = $\ln{|\tan{\frac{x}{2}}|}$ = $ln | csc x - cot x |$ (because $\tan{\frac{x}{2}} = \csc{x} - \cot{x}$ ).

Proving that $\tan{\frac{x}{2}} = \csc{x} - \cot{x}$

$\tan{\frac{x}{2}} = \csc{x} - \cot{x}$ can also be written as $tan x = csc2 x - cot2 x$ .

$csc2 x - cot2 x$ = $\frac{1}{\sin{2x}} - \frac{\cos{2x}}{\sin{2x}}$ = $\frac{1-\cos{2x}}{\sin{2x}}$ = $\frac{1-(1-2\sin^2{x})}{2\cos{x}\sin{x}}$ = $\frac{2\sin{x}\sin{x}}{2\cos{x}\sin{x}}$ = $\frac{\sin{x}}{\cos{x}}$ = $tan x$

Retrieved from "http://en.wikipedia.org/wiki/User:Thatha"

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