Solve tan(x)+cos(x)=1/2 Is it possible (not numerically) to find the $x$ such as:
$$
tan(x)+cos(x)=1/2
$$
?
All my tries finishes in a 4 degree polynomial. By example, calling c = cos(x):
$$
\frac{\sqrt{1-c^2}}{c}+c=\frac{1}{2}
$$
$$
\sqrt{1-c^2}+c^2=\frac{1}{2}c
$$
$$
1-c^2=c^2(\frac{1}{2}-c)^2=c^2(\frac{1}{4}-c+c^2)
$$
$$
c^4-c^3+\frac{5}{4}c^2-1=0
$$

If we put $t=x/2$ then we get $${2\tan t\over 1-\tan ^2t} +2\cos ^2 t -1={1\over 2}$$ 
Let $y= \tan t$. Since $\cos ^2t = {1\over 1+y^2}$ we get $$3y^4+4y^3-4y^2+4y+1=0$$
which I'm not sure if any helps. :(
