What is cot A + cosec A equal to?

Concept:

cos 2A = cos² A - sin² A = 2 cos² A - 1

sin 2A = 2 sin A × cos A

Calculation:

\(\Rightarrow \cot A+cosec~A=\frac{\cos A}{\sin A}+\frac{1}{\sin A}\)

\(\Rightarrow \cot A+cosec~A=\frac{1+\cos A}{\sin A}\)

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

\(\Rightarrow \cot A+cosec~A=\frac{1+\cos A}{\sin A}\)

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

∴ The value of cot A + cosec A is \(\cot \left( \frac{A}{2} \right)\)