# 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. Commented Oct 12, 2015 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
Commented Oct 12, 2015 at 19:36
• (...together with the solutions obtained by exploiting the symmetry of the tangent function.) Commented Oct 12, 2015 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)? Commented Oct 12, 2015 at 19:38
• Are both answers correct then? Commented Oct 12, 2015 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 $$2 \sin x= \cos x$$ implies $$t^2+4t-1=0$$ from which $$\tan \frac x2=2\pm\sqrt{5}$$. Hence the answer of Wolfram.