If cos cos = a and sin sin = b , then the value of ^2 in terms of a and b is?

Answer: Option C

$$\begin{array}{rl}& \frac{\text{cos }\alpha }{\text{cos }\beta }=a\\ & \Rightarrow \cos \alpha =a\cos \beta \\ & \text{On squaring both sides}\\ & {\cos }^2\alpha =a^2{\cos }^2\beta \\ & \Rightarrow 1-{\sin }^2\alpha =a^2\left(1-{\sin }^2\beta \right)....(i)\\ & \text{Again, }\sin \alpha =b\sin \beta \\ & \text{Squaring both sides}\\ & \Rightarrow {\sin }^2\alpha =b^2{\sin }^2\beta \\ & \text{Put the value of }{\sin }^2\alpha \text{ in equation (i)}\\ & \Rightarrow \text{1}-b^2{\sin }^2\beta =a^2-a^{2}si{n}^2\beta \\ & \Rightarrow a^2-1=a^{2}si{n}^2\beta -b^{2}si{n}^2\beta \\ & \Rightarrow a^2-1=si{n}^2\beta \left(a^2-b^2\right)\\ & \Rightarrow si{n}^2\beta =\frac{a^2-1}{a^2-b^2}\end{array}$$