How to Simplify $-\frac14\sin\frac x4 + \frac{\sqrt{3}}4\cos\frac x4 = 0$? A problem requires finding the value of x, such that 
$$-\frac14\sin\frac x4 + \frac{\sqrt{3}}4\cos\frac x4 = 0$$
The site reduces the problem to $\tan(\frac x4) = \sqrt{3}$.
How one should do this? 

Method 1
$$\begin{align}
-\frac14\sin\frac x4 + \frac{\sqrt{3}}4\cos\frac x4 &= 0 \\
-\frac14\sin\frac x4 &= - \frac{\sqrt{3}}4\cos\frac x4 & \text{Take cos to the other side}\\
\sin\frac x4 &= \sqrt{3} \cos \frac x4 & \text{Multiply by -4} \\
\tan \frac x4 &= \sqrt{3} & \text{Devide by cos}
\end{align}$$
But in this method you should take care that $\cos \frac x4 \ne 0$ as you devided by it at the last step. But happily $\cos \frac x4 = 0$ will never happen! (Why?)
Method 2
$$\begin{align}
-\frac14\sin\frac x4 + \frac{\sqrt{3}}4\cos\frac x4 &= 0 \\
-\frac12 \left(\frac 12 \sin\frac x4 - \frac{\sqrt{3}}2\cos\frac x4 \right) &= 0  & \text{Factor $-\frac {1} {2}$}\\
-\frac12 \left(\cos \frac \pi 3 \sin\frac x4 - \sin \frac \pi 3 \cos\frac x4 \right) &= 0 & \text{Note that $\cos \frac \pi 3 = \frac 1 2$ and $\sin \frac \pi 3 = \frac {\sqrt{3}} {2}$} \\
-\frac{1}{2} \sin \left( \frac x 4 - \frac \pi 3 \right) &= 0 & \text{Use summation formula for sin} \\
\sin \left( \frac x 4 - \frac \pi 3 \right) &=0
\end{align}$$
You can take a look at this post for a more general case of method $2$.
Solution of the Equation
Using either method $1$ or method $2$ one will conclude that
$$x=\frac{4\pi}{3}+4n\pi, \qquad \qquad n=0,\pm1,\pm2,...$$
