What is the value of $$\frac{{\left[ {\sin \left( {90 - A} \right) + \cos \left( {180 - 2A} \right)} \right]}}{{\left[ {\cos \left( {90 - 2A} \right) + \sin \left( {180 - A} \right)} \right]}}?$$

Solution (By Examveda Team)

$$\eqalign{ & \frac{{\left[ {\sin \left( {90 - A} \right) + \cos \left( {180 - 2A} \right)} \right]}}{{\left[ {\cos \left( {90 - 2A} \right) + \sin \left( {180 - A} \right)} \right]}} \cr & \Rightarrow \frac{{\cos A + \left( { - \cos 2A} \right)}}{{\sin 2A + \sin A}} \cr & \Rightarrow \frac{{\cos A - \cos 2A}}{{\sin 2A + \sin A}} \cr & \Rightarrow \frac{{ - 2\sin \left( {\frac{{A + 2A}}{2}} \right) \times \sin \left( {\frac{{A - 2A}}{2}} \right)}}{{2\sin \left( {\frac{{A + 2A}}{2}} \right) \times \cos \left( {\frac{{2A - A}}{2}} \right)}} \cr & \Rightarrow \frac{{ - \sin \left( {\frac{{A - 2A}}{2}} \right)}}{{\cos \left( {\frac{{2A - A}}{2}} \right)}} \cr & \Rightarrow \frac{{\sin \left( {\frac{{2A - A}}{2}} \right)}}{{\cos \left( {\frac{{2A - A}}{2}} \right)}} \cr & \Rightarrow \frac{{\sin \left( {\frac{A}{2}} \right)}}{{\cos \left( {\frac{A}{2}} \right)}} \cr & \Rightarrow \tan \left( {\frac{A}{2}} \right) \cr} $$