Concept:

• \({\cot ^{ - 1}}\left( {\cot x} \right) = x\)
• \(\cot \left( {x + y} \right) = \frac{{\cot x\cot y - 1}}{{\cot y + \cot x}}\)

Calculation:

The expression \({\cot ^{ - 1}}\left( {\frac{{1 - \sin x}}{{\cos x}}} \right)\) can be written as follows:

\({\cot ^{ - 1}}\left( {\frac{{1 - \sin x}}{{\cos x}}} \right) = {\cot ^{ - 1}}\left( {\frac{{{{\cos }^2}\left( {x/2} \right) + {{\sin }^2}\left( {x/2} \right) - 2\sin \left( {x/2} \right)\cos \left( {x/2} \right)}}{{{{\cos }^2}\left( {x/2} \right) - {{\sin }^2}\left( {x/2} \right)}}} \right)\)

\( = {\cot ^{ - 1}}\left( {\frac{{{{\left( {\cos \left( {x/2} \right) - \sin \left( {x/2} \right)} \right)}^2}}}{{\left( {\cos \left( {x/2} \right) - \sin \left( {x/2} \right)} \right)\left( {\cos \left( {x/2} \right) + \sin \left( {x/2} \right)} \right)}}} \right)\)

\(= {\cot ^{ - 1}}\left( {\frac{{\left( {\cos \left( {x/2} \right) - \sin \left( {x/2} \right)} \right)}}{{\left( {\cos \left( {x/2} \right) + \sin \left( {x/2} \right)} \right)}}} \right)\)

Dividing the numerator and denominator of RHS by sin(x/2) , we get

\({\cot ^{ - 1}}\left( {\frac{{1 - \sin x}}{{\cos x}}} \right) = {\cot ^{ - 1}}\left( {\frac{{\left( {cot(\frac{x}{2})\ cot \frac{\pi}{4} - \ 1} \right)}}{{\left( {cot(\frac{\pi}{4} + \cot \left( {\frac{x}{2}} \right)} \right)}}} \right)\)

As we know that, \(\cot \left( {x + y} \right) = \frac{{\cot x\cot y - 1}}{{\cot y + \cot x}}\)

\(= {\cot ^{ - 1}}\left( {\cot \left( {\frac{\pi }{4} + \frac{x}{2}} \right)} \right)\)

\(= \frac{\pi }{4} + \frac{x}{2}\)