Question 633154
<font face="Times New Roman" size="+2">


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sin(x)\ -\ \sin(2x)\ +\ 2\cos(x)\ -\ 1\ =\ 0]


Use


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sin(2\varphi)\ \equiv\ 2\sin(\varphi)\cos(\varphi)]


To write:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sin(x)\ -\ 2\sin(x)\cos(x)\ +\ 2\cos(x)\ -\ 1\ =\ 0]


Collect the *[tex \LARGE \sin] terms in the LHS and the rest in the RHS:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sin(x)\ -\ 2\sin(x)\cos(x)\ =\ 1\ -\ 2\cos(x)]


Factor out *[tex \LARGE \sin(x)]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sin(x)\left(1\ -\ 2\cos(x)\right)\ =\ 1\ -\ 2\cos(x)]


Multiply both sides by


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{1}{1\ -\ \cos(x)}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sin(x)\ =\ 1]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{\pi}{4}\ +\ 2k\pi,\ \text{where}\ k\ \in\ \mathbb{Z}]


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
<div style="text-align:center"><a href="http://outcampaign.org/" target="_blank"><img src="http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png" border="0" alt="The Out Campaign: Scarlet Letter of Atheism" width="143" height="122" /></a></div>
</font>