In a triangle ABC, a – 2b + c = 0. The value of \(\cot \left( {\frac{A}{2}} \right)\cot \left( {\frac{C}{2}} \right)\) is
3
Concept:
Properties of Triangle:
If the three sides a, b, and c are the sides of a triangle,\({\rm{\Delta }} = {\rm{\;}}\sqrt {{\rm{s}}\left( {{\rm{s}} - {\rm{a}}} \right)\left( {{\rm{s}} - {\rm{b}}} \right)\left( {{\rm{s}} - {\rm{c}}} \right)}\)where \({\rm{s}} = \frac{{{\rm{a}} + {\rm{b}} + {\rm{c}}}}{2}\) where
S = semi-perimeter of a triangle
Semi-angle formulas of a triangle:
In Δ ABC
- \(\sin \left( {\frac{{\rm{A}}}{2}} \right) = {\rm{\;}}\sqrt {\frac{{({\rm{s}} - {\rm{b}}(\left( {{\rm{s}} - {\rm{c}}} \right)}}{{{\rm{bc}}}}}\)
- \(\cos \left( {\frac{{\rm{A}}}{2}} \right) = {\rm{\;}}\sqrt {\frac{{{\rm{s}}\left( {{\rm{s}} - {\rm{a}}} \right)}}{{{\rm{bc}}}}}\)
- \(\tan \left( {\frac{{\rm{A}}}{2}} \right) = {\rm{\;}}\sqrt {\frac{{\left( {{\rm{s}} - {\rm{b}}} \right)\left( {{\rm{s}} - {\rm{c}}} \right)}}{{{\rm{s}}\left( {{\rm{s}} - {\rm{a}}} \right)}}}\)
Calculation:
It is given that, a – 2b + c = 0
⇒ a + c = 2b
As we know that the semi-perimeter S of a triangle = (a + b + c)/2
⇒ s = (2b + b) /2 = 3b/2
As we know that
\({\rm{Cot\;}}\left( {\frac{{\rm{A}}}{2}} \right) = \sqrt {\frac{{{\rm{s}}\left( {{\rm{s}} - {\rm{a\;}}} \right)}}{{\left( {{\rm{s}} - {\rm{b}}} \right)\left( {{\rm{s}} - {\rm{c}}} \right)}}} ,\cot \left( {\frac{{\rm{c}}}{2}} \right) = {\rm{\;}}\sqrt {\frac{{{\rm{s}}\left( {{\rm{s}} - {\rm{c}}} \right)}}{{\left( {{\rm{s}} - {\rm{a}}} \right)\left( {{\rm{s}} - {\rm{b}}} \right)}}}\)
\(\Rightarrow \cot \left( {\frac{{\rm{A}}}{2}} \right)\cot \left( {\frac{{\rm{c}}}{2}} \right) = {\rm{\;}}\sqrt {\frac{{{\rm{s}}\left( {{\rm{s}} - {\rm{a\;}}} \right)}}{{\left( {{\rm{s}} - {\rm{b}}} \right)\left( {{\rm{s}} - {\rm{c}}} \right)}}} \sqrt {\frac{{{\rm{s}}\left( {{\rm{s}} - {\rm{c}}} \right)}}{{\left( {{\rm{s}} - {\rm{a}}} \right)\left( {{\rm{s}} - {\rm{b}}} \right)}}}\)
\(\Rightarrow \cot \left( {\frac{{\rm{A}}}{2}} \right)\cot \left( {\frac{{\rm{c}}}{2}} \right) = {\rm{\;}}\sqrt {\frac{{{{\rm{s}}^2}}}{{{{\left( {{\rm{s}} - {\rm{b}}} \right)}^2}}}}\)
\(\Rightarrow \cot \left( {\frac{{\rm{A}}}{2}} \right)\cot \left( {\frac{{\rm{c}}}{2}} \right) = \frac{{\rm{s}}}{{\left( {{\rm{s}} - {\rm{b}}} \right)}}\)
\(\Rightarrow \cot \left( {\frac{{\rm{A}}}{2}} \right)\cot \left( {\frac{{\rm{c}}}{2}} \right) = \frac{{\frac{{3{\rm{b}}}}{2}}}{{\frac{{3{\rm{b}}}}{2} - {\rm{b}}}} = 3\)
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