Question
Download Solution PDFWhat is \(\cot \left( {\frac{A}{2}} \right) - \tan \left( {\frac{A}{2}} \right)\) equal to?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
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
\(\Rightarrow \cot \left( {\frac{A}{2}} \right) - \tan \left( {\frac{A}{2}} \right) = \frac{{\cos A}}{{\frac{1}{2} \times 2 \times \sin \left( {\frac{A}{2}} \right) \times \cos \left( {\frac{A}{2}} \right)\;}} = \;2\frac{{\cos A}}{{\sin A}} = 2\cot A\)
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