Correct Answer - Option 2 :
\(\cot \left( \frac{A}{2} \right)\)
Concept:
cos 2A = cos2 A - sin2 A = 2 cos2 A - 1
sin 2A = 2 sin A × cos A
Calculation:
\(\Rightarrow \cot A+cosec~A=\frac{\cos A}{\sin A}+\frac{1}{\sin A}=\frac{1+\cos A}{\sin A}\)
As we know that, cos 2A = cos2 A - sin2 A = 2 cos2 A - 1
\(\Rightarrow \cot A+cosec~A=\frac{1+\cos A}{\sin A}=\frac{2\times {{\cos }^{2}}\left( \frac{A}{2} \right)}{\sin A}\)
As we know that, sin 2A = 2 sin A × cos A
\(\Rightarrow \cot A+cosec~A=\frac{1+\cos A}{\sin A}=\frac{2\times {{\cos }^{2}}\left( \frac{A}{2} \right)}{\sin A}=\frac{2\times {{\cos }^{2}}\left( \frac{A}{2} \right)}{2\times \cos \left( \frac{A}{2} \right)\times \sin \left( \frac{A}{2} \right)}=\cot \left( \frac{A}{2} \right)\)