# How to solve equations like $2 \sin(x) = \cos(x)$

My mathbook tells me that it isn't possible to solve this:

$$2 \sin(x) = \cos(x)$$

But Wolfram Alpha gives the following answer:

$$x = 2\cdot\left(\pi n-\tan^{-1}(2\pm\sqrt{5})\right)$$

Is it possible to do this, without the help of a calculator?

• Well you can find $\tan x$ easily enough the way you've written it - have you got the question right. – Mark Bennet Oct 12 '15 at 19:34
• The way you wrote it would just be $\tan(x)=\frac 1 2$, so your answer would be $\arctan (\frac 1 2)$ – Alan Oct 12 '15 at 19:36
• (...together with the solutions obtained by exploiting the symmetry of the tangent function.) – Travis Oct 12 '15 at 19:36
• Is the question, then, how does one show that WolframAlpha's expression is the solution (or equivalently, equivalent to the solution described here in the comments)? – Travis Oct 12 '15 at 19:38
• Are both answers correct then? – Adnan Oct 12 '15 at 19:44

One way can be using tan$\frac x2=t$ so sin x=$\frac{2t}{1+t^2}$ and cos x=$\frac{1-t^2}{1+t^2}$.

Here 2sin x= cos x implies $t^2+4t-1=0$ from wich tan $\frac x2=2\pm\sqrt{5}$.Hence the answer of Wolphram.