Trigonometry solutions without calculator $\cos x =\frac{\sqrt{5} +1}4$ where x is in radians  results in one solution of $x=\frac\pi5$, how do I prove this without the use of a calculator.
similarly if $\tan x = 2 +\sqrt{3}$ and $x$ is $0<x<\frac\pi2$ find without the use of a calculator the exact value of $x$
 A: In general, there isn't an exact analytic expression for the inverse trigonometric functions of an arbitrary radical. 
Only a handful of small rational multiples of $\pi$ have trig functions that can be expressed using simple rational number and radicals. Sure, you can extend those a little with half-angle formula & addition and subtraction rules, but you soon run into horrible nested radicals, and for multiples of $\pi/n$ for $n>5$ you're generally dealing with equations that don't have solutions that can be expressed as radicals, as proven by the Abel–Ruffini theorem. 
The same logic applies going in the reverse direction. Given a trigonometric ratio written as some arbitrary radical expression there generally isn't an exact analytic expression involving multiples &/or radicals of $\pi$ which is the inverse function of that ratio.
A: Hint:
$$2+\sqrt3=\csc30+\cot30=\cdots=\cot15$$
If $4\cos x=\sqrt5+1,5=(4\cos x-1)^2$
$$4\cos^2x-2\cos x-1=0$$
Now see http://mathforum.org/library/drmath/view/54090.html
