What is \(\cot \left( {\frac{A}{2}} \right) - \tan \left( {\frac{A}{2}} \right)\) equal to?

Concept:

cos 2A = cos² A – sin² A

sin 2A = 2 sin A × cos A

Calculation:

\(\Rightarrow \cot \left( {\frac{A}{2}} \right) - \tan \left( {\frac{A}{2}} \right) = \frac{{\cos \left( {\frac{A}{2}} \right)}}{{\sin \left( {\frac{A}{2}} \right)}} - \frac{{\sin \left( {\frac{A}{2}} \right)}}{{\cos \left( {\frac{A}{2}} \right)}} = \frac{{{{\cos }^2}\left( {\frac{A}{2}} \right) - {{\sin }^2}\left( {\frac{A}{2}} \right)}}{{\sin \left( {\frac{A}{2}} \right) \times \cos \left( {\frac{A}{2}} \right)}}\)

As we know that, cos 2A = cos² A – sin² A

\(\Rightarrow \cot \left( {\frac{A}{2}} \right) - \tan \left( {\frac{A}{2}} \right) = \frac{{{{\cos }^2}\left( {\frac{A}{2}} \right) - {{\sin }^2}\left( {\frac{A}{2}} \right)}}{{\sin \left( {\frac{A}{2}} \right) \times \cos \left( {\frac{A}{2}} \right)}} = \frac{{\cos A}}{{\frac{1}{2} \times 2 \times \sin \left( {\frac{A}{2}} \right) \times \cos \left( {\frac{A}{2}} \right)\;}}\)

As we know that, sin 2A = 2 sin A × cos A

\(\Rightarrow \cot \left( {\frac{A}{2}} \right) - \tan \left( {\frac{A}{2}} \right) = \frac{{\cos A}}{{\frac{1}{2} \times 2 \times \sin \left( {\frac{A}{2}} \right) \times \cos \left( {\frac{A}{2}} \right)\;}} = \;2\frac{{\cos A}}{{\sin A}} = 2\cot A\)