I think there must be multiple approaches to this proof. Below is what I did. The whole idea is to find a connection (equation) between the length and the given angle .
Please see the attached the figure, I draw a line from the vertex of and parallel to lower side of the rectangle. According to your given information, we have , and .
In right triangle , we have . Realize that . So we have .
In right triangle , we have . Substitute obtained earlier, we have