What is the value of [ ( 90 - A ) + ( 180 - 2A ) ] [ ( 90 - 2A ) + ( 180 - A ) ]?

Answer: Option C

$$\begin{array}{rl}& \frac{\left[\sin \left(90-A\right)+\cos \left(180-2A\right)\right]}{\left[\cos \left(90-2A\right)+\sin \left(180-A\right)\right]}\\ & \Rightarrow \frac{\cos A+\left(-\cos 2A\right)}{\sin 2A+\sin A}\\ & \Rightarrow \frac{\cos A-\cos 2A}{\sin 2A+\sin A}\\ & \Rightarrow \frac{-2\sin \left(\frac{A+2A}{2}\right)\times \sin \left(\frac{A-2A}{2}\right)}{2\sin \left(\frac{A+2A}{2}\right)\times \cos \left(\frac{2A-A}{2}\right)}\\ & \Rightarrow \frac{-\sin \left(\frac{A-2A}{2}\right)}{\cos \left(\frac{2A-A}{2}\right)}\\ & \Rightarrow \frac{\sin \left(\frac{2A-A}{2}\right)}{\cos \left(\frac{2A-A}{2}\right)}\\ & \Rightarrow \frac{\sin \left(\frac{A}{2}\right)}{\cos \left(\frac{A}{2}\right)}\\ & \Rightarrow \tan \left(\frac{A}{2}\right)\end{array}$$