Two men on either side of a cliff, $60 \mathrm{m}$ high, observe the angles of elevation of the top of the cliff to be $45°$ and $60°$ respectively. The distance between two men is $k\left(3+\sqrt{3}\right) \mathrm{m}$. Find $k$.

The string of a kite is $100 \mathrm{m}$ long and it makes an angle of $60°$ with the horizontal. The height of the kite is $k\sqrt{3}m$, then find the value of $k$.

In the figure, $△\mathrm{ABC}$ is a right-angled triangle. $\mathrm{D}$ is the mid-point of $\mathrm{BC}$ Show that $\frac{\mathrm{tan\theta }}{\mathrm{tan\varphi }}=\frac{1}{2}$

The angle of depression of two ships from the top of a lighthouse are $45°$ and $30°$ towards east. If the ships are $200$ meters apart, find the height of the light house.