What is the value of [ ( 90^ - A ) + ( 90^ - A ) ]^2 [ 2 ^2 ( 90^ - 2A ) ]?

Answer: Option C

$$\begin{array}{rl}& \frac{{\left[\tan \left({90}^{\circ }-A\right)+\cot \left({90}^{\circ }-A\right)\right]}^2}{\left[2{\sec }^2\left({90}^{\circ }-2A\right)\right]}\\ & \text{By putting }A=45^{\circ }\\ & \Rightarrow \frac{{\left[\tan \left({90}^{\circ }-{45}^{\circ }\right)+\cot \left({90}^{\circ }-{45}^{\circ }\right)\right]}^2}{\left[2{\sec }^2\left({90}^{\circ }-2\times {45}^{\circ }\right)\right]}\\ & \Rightarrow \frac{{\left[\tan {45}^{\circ }+\cot {45}^{\circ }\right]}^2}{2{\sec }^2\left({90}^{\circ }-{90}^{\circ }\right)}\\ & \Rightarrow \frac{{\left[1+1\right]}^2}{2{\sec }^2{0}^{\circ }}\\ & \Rightarrow \frac{4}{2\times 1}\\ & \Rightarrow 2\end{array}$$