How to find angle x without calculator? 
How to find  angle x without using calculator? 
By using the trigonometric ratio formula, we can evaluate x but  in this case calculator is necessary. 
 A: We have $$\frac{AD}{\sin x}=\frac2{\sin(120^\circ-x)}$$ and $$\frac{AD}{\sin45^\circ}=\frac1{\sin15^\circ}$$
On division, $$\frac{\sin(120^\circ-x)}{\sin x}=\frac{2\sin15^\circ}{\sin45^\circ}$$
Now $\displaystyle\sin45^\circ=3\sin15^\circ-4\sin^315^\circ=\sin15^\circ(3-4\sin^215^\circ)=\sin15^\circ[3-2(1-\cos30^\circ)]$
$\displaystyle\implies\frac{\sin45^\circ}{\sin15^\circ}=1+2\cos30^\circ=1+\sqrt3=\frac2{\sqrt3-1}$
$$\implies\sin120^\circ\cot x-\cos120^\circ=\sqrt3-1$$
$$\implies\frac{\sqrt3}2\cot x=\sqrt3-1-\frac12$$
$$\iff\cot x=2-\sqrt3=\cot75^\circ$$
as $$\cos2x=\frac{1-\tan^2x}{1+\tan^2x}=\frac{\cot^2x-1}{\cot^2x+1}=\cdots=-\frac{\sqrt3}2$$
A: The fastest way out here would be to use the $m$-$n$ $\cot$ theorem. If you are not familiar with it, here it is:

If $D$ be the point on the side $BC$ of a triangle $\triangle ABC$ which divides the side $BC$ in the ratio $m$ : $n$ , then with respect to the figure given below, we have:

$$(m+n)\cot\theta=m\cot\alpha-n\cot\beta\tag{i}$$ $$(m+n)\cot\theta=n\cot B-m\cot C\tag{ii}$$

Using (ii) in your question directly gives $\cot x=2-\sqrt 3$ and hence $\color{red}{x=75^{\circ}}$
