<s>[INST] Find $AC$ WITHOUT using Cosine law... 
In the following figure find side $AC$ WITHOUT using Cosine law:
  

There is a simple solution involving Cosine law,but is it possible to calculate $AC$ without using it?? 
 [/INST] Let $BC=x$. Then $AC=\sqrt{1+x^2}$. Use the law of sines in triangle BCD to obtain
$$\frac{x}{\sin\angle D}=\frac{1}{\sin 30^\circ}=2.$$
Notice that $\angle ABD=120^\circ$, whose supplementary angle is $60^\circ$. So we have
$$ AB\times\sin 60^\circ=AD\times\sin\angle D.$$
We may plug in numbers to obtain
$$(1+\sqrt{1+x^2})\,\frac{x}{2}=\frac{\sqrt{3}}{2}.$$
Here's the tricky part. In terms of $y=\sqrt{1+x^2}$, the equation becomes
\begin{align}
(1+y)\sqrt{y^2-1}&=\sqrt{3},\\
y^4+2y^3-2y-1&=3,\\
(y^3-2)(y+2)&=0.
\end{align}
The answer is $AC=2^{1/3}$.
</s>