Angle $\alpha$ is divided into parts such that the ratio of tangents is $k$ and the difference of parts is $x$. Show $\sin x=(k-1)/(k+1)\sin\alpha$.

Question

Let angle $\alpha$ be divided into two parts such that the ratio of their tangents is $k$ and the difference between the parts is $x$. Then show that

$$\sin x = \frac{k-1}{k+1}\sin \alpha$$

I know the result can be derived easily using Componendo and Dividendo but is there a more intuitive way for the same?