cos sin = n and cos cos = m, then the value of co s^2 is?

Answer: Option D

$$\begin{array}{rl}& \text{Given,}\\ & n=\frac{\text{cos }\alpha }{\text{sin }\beta }\text{,}\text{m}=\frac{\text{cos }\alpha }{\text{cos }\beta }\\ & \Rightarrow \text{cos }\alpha =n\text{ sin }\beta ,and\cos \alpha =m\cos \beta \\ & \Rightarrow \text{co}{\text{s}}^2\alpha =n^2\text{ si}{\text{n}}^2\beta .....(i)\\ & \Rightarrow \text{co}{\text{s}}^2\alpha =m^2\text{ co}{\text{s}}^2\beta .....(ii)\\ & equation(i)=(ii)\\ & \Rightarrow n^2\text{ si}{\text{n}}^2\beta =m^2\text{ co}{\text{s}}^2\beta \\ & \Rightarrow n^2\left(\text{1}-\text{ co}{\text{s}}^2\beta \right)=m^2\text{ co}{\text{s}}^2\beta \\ & \Rightarrow n^2-n^2{\cos }^2\beta =m^2{\cos }^2\beta \\ & \Rightarrow n^2=m^2{\cos }^2\beta +n^2{\cos }^2\beta \\ & \Rightarrow n^2={\cos }^2\beta \left(m^2+n^2\right)\\ & \Rightarrow {\cos }^2\beta =\frac{n^2}{m^2+n^2}\end{array}$$